The Vigenere Cipher
Author: R. Morelli
One of the main problems with simple substitution ciphers is that they are so vulnerable to frequency analysis. Given a sufficiently large ciphertext, it can easily be broken by mapping the frequency of its letters to the know frequencies of, say, English text. Therefore, to make ciphers more secure, cryptographers have long been interested in developing enciphering techniques that are immune to frequency analysis. One of the most common approaches is to suppress the normal frequency data by using more than one alphabet to encrypt the message. A polyalphabetic substitution cipher involves the use of two or more cipher alphabets. Instead of there being a one-to-one relationship between each letter and its substitute, there is a one-to-many relationship between each letter and its substitutes.
The Vigenere Tableau
The Vigenere Cipher, proposed by Blaise de Vigenere from the court of Henry III of France in the sixteenth century, is a polyalphabetic substitution based on the following tableau:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
A A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
B B C D E F G H I J K L M N O P Q R S T U V W X Y Z A
C C D E F G H I J K L M N O P Q R S T U V W X Y Z A B
D D E F G H I J K L M N O P Q R S T U V W X Y Z A B C
E E F G H I J K L M N O P Q R S T U V W X Y Z A B C D
F F G H I J K L M N O P Q R S T U V W X Y Z A B C D E
G G H I J K L M N O P Q R S T U V W X Y Z A B C D E F
H H I J K L M N O P Q R S T U V W X Y Z A B C D E F G
I I J K L M N O P Q R S T U V W X Y Z A B C D E F G H
J J K L M N O P Q R S T U V W X Y Z A B C D E F G H I
K K L M N O P Q R S T U V W X Y Z A B C D E F G H I J
L L M N O P Q R S T U V W X Y Z A B C D E F G H I J K
M M N O P Q R S T U V W X Y Z A B C D E F G H I J K L
N N O P Q R S T U V W X Y Z A B C D E F G H I J K L M
O O P Q R S T U V W X Y Z A B C D E F G H I J K L M N
P P Q R S T U V W X Y Z A B C D E F G H I J K L M N O
Q Q R S T U V W X Y Z A B C D E F G H I J K L M N O P
R R S T U V W X Y Z A B C D E F G H I J K L M N O P Q
S S T U V W X Y Z A B C D E F G H I J K L M N O P Q R
T T U V W X Y Z A B C D E F G H I J K L M N O P Q R S
U U V W X Y Z A B C D E F G H I J K L M N O P Q R S T
V V W X Y Z A B C D E F G H I J K L M N O P Q R S T U
W W X Y Z A B C D E F G H I J K L M N O P Q R S T U V
X X Y Z A B C D E F G H I J K L M N O P Q R S T U V W
Y Y Z A B C D E F G H I J K L M N O P Q R S T U V W X
Z Z A B C D E F G H I J K L M N O P Q R S T U V W X Y
Note that each row of the table corresponds to a Caesar Cipher. The first row is a shift of 0; the second is a shift of 1; and the last is a shift of 25.
The Vigenere cipher uses this table together with a keyword to encipher a message. For example, suppose we wish to encipher the plaintext message:
TO BE OR NOT TO BE THAT IS THE QUESTION
using the keyword RELATIONS. We begin by writing the keyword, repeated as many times as necessary, above the plaintext message. To derive the ciphertext using the tableau, for each letter in the plaintext, one finds the intersection of the row given by the corresponding keyword letter and the column given by the plaintext letter itself to pick out the ciphertext letter.
Keyword: RELAT IONSR ELATI ONSRE LATIO NSREL
Plaintext: TOBEO RNOTT OBETH ATIST HEQUE STION
Ciphertext: KSMEH ZBBLK SMEMP OGAJX SEJCS FLZSY
Decipherment of an encrypted message is equally straightforward. One writes the keyword repeatedly above the message:
Keyword: RELAT IONSR ELATI ONSRE LATIO NSREL
Ciphertext: KSMEH ZBBLK SMEMP OGAJX SEJCS FLZSY
Plaintext: TOBEO RNOTT OBETH ATIST HEQUE STION
This time one uses the keyword letter to pick a column of the table and then traces down the column to the row containing the ciphertext letter. The index of that row is the plaintext letter.
The strength of the Vigenere cipher against frequency analysis can be seen by examining the above ciphertext. Note that there are 7 'T's in the plaintext message and that they have been encrypted by 'H,' 'L,' 'K,' 'M,' 'G,' 'X,' and 'L' respectively. This successfully masks the frequency characteristics of the English 'T.' One way of looking at this is to notice that each letter of our keyword RELATIONS picks out 1 of the 26 possible substitution alphabets given in the Vigenere tableau. Thus, any message encrypted by a Vigenere cipher is a collection of as many simple substitution ciphers as there are letters in the keyword.
Although the Vigenere cipher has all the features of a useful field cipher -- i.e., easily transportable key and tableau, requires no special apparatus, easy to apply, etc. -- it did not catch on its day. A variation of it, known as the Gronsfeld cipher , did catch on in Germany and was widely used in Central Europe. The Gronsfeld variant used the digits of a keynumber instead of a the letters of keyword, but remained unchanged in all other respects. So in fact the Gronsfeld is a weaker technique than Vigenere since it only uses 10 substitute alphabets (one per digit 0..9) instead of the 26 used by Vigenere.
Cryptanalyzing the Vigenere Cipher: The Kasiski/Kerckhoff Method
Vigenere-like substitution ciphers were regarded by many as practically unbreakable for 300 years. In 1863, a Prussian major named Kasiski proposed a method for breaking a Vigenere cipher that consisted of finding the length of the keyword and then dividing the message into that many simple substitution cryptograms. Frequency analysis could then be used to solve the resulting simple substitutions.
Kasiski's technique for finding the length of the keyword was based on measuring the distance between repeated bigrams in the ciphertext. Note that in the above cryptogram the plaintext bigram 'TO' occurs twice in the message at position 0 and 9 and in both cases it lines up perfectly with the first two letters of the keyword. Because of this it produces the same ciphertext bigram, 'KS.' The same can be said of plaintext 'BE' which occurs twice starting at positions 2 and 11, and also is encrypted with the same ciphertext bigram, 'ME.' In fact, any message encrypted with a Vigenere cipher will produce many such repeated bigrams. Although not every repeated bigram will be the result of the encryption of the same plaintext bigram, many will, and this provides the basis for breaking the cipher. By measuring and factoring the distances between recurring bigrams -- in this case the distance is 9 -- Kasiski was able to guess the length of the keyword. For this example,
Location: 01234 56789 01234 56789 01234 56789
Keyword: RELAT IONSR ELATI ONSRE LATIO NSREL
Plaintext: TOBEO RNOTT OBETH ATIST HEQUE STION
Ciphertext: KSMEH ZBBLK SMEMP OGAJX SEJCS FLZSY
the Kasiski method would create something like the following list:
Repeated Bigram Location Distance Factors
KS 9 9 3, 9
SM 10 9 3, 9
ME 11 9 3, 9
...
Factoring the distances between repeated bigrams is a way of identifying possible keyword lengths. Those factors that occur most frequently will be the best candidates for the length of the keyword. Note that in this example since 3 is also a factor of 9 (and any of its multiples) both 3 and 9 would be reasonable candidates for keyword length. Although in this example we don't have a clear favorite, we've narrowed down the possibilities to a very small list. Note also that if a longer ciphertext were encrypted with the same keyword ('RELATIONS'), we would expect to find repeated bigrams at multiples of 9 -- i.e., 18, 27, 81, etc. These would also have 3 as a factor. Kasiski's important contribution is to note this phenomenon of repeated bigrams and propose a method -- factoring of distances -- to analyze it.
Once the length of the keyword is known, the ciphertext can be broken up into that many simple substitution cryptograms. That is, for a keyword of length 9, every 9-th letter in the ciphertext was encrypted with the same keyword letter. Given the structure of the Vigenere tableau, this is equivalent to using 9 distinct simple substitution ciphers, each of which was derived from 1 of the 26 possible Caesar shifts given in the tableau. The pure Kasiski method proceeds by analyzing these simple substitution cryptograms using frequency analysis and the other standard techniques.
A variant of this method, proposed by the French cryptographer Kerckhoff, is based on discovering the keyword itself and then using it to decipher the cryptogram. In Kerckhoff's method, after the message has been separated into several columns, corresponding to the simple substitution cryptograms, one tallies the frequencies in each column and then uses frequency and logical analysis to construct the key. For example, suppose the most frequent letter in the first column is 'K'. We would hypothesize that 'K' corresponds to the English 'E'. If we consult the Vigenere tableau at this point, we can see that if English 'E' were enciphered into 'K' then row G of the table must have been the alphabet used for the first letter of the keyword. This implies that the first letter of the keyword is 'G'.
The problem with this "manual" approach is that for short messages there are often several good candidates for English 'E' in each column. This requires the testing of multiple hypotheses, which can get quite tedious and involved. Therefore we need a more sensitive test to discover the alphabet used by each letter of the keyword.
Recalling that each row of the Vigenere tableau is one of the 26 Caesar shifts, we can use the chi-square test to determine which of the 26 possible shifts was used for each letter of the keyword. This modern day version of the Kerckhoff method turns out to be very effective. And this is the algorithm that is used in CryptoToolJ's Vigenere Analyzer.
http://www.cs.trincoll.edu/~crypto/historical/vigenere.html
Sunday, November 7, 2010
Maryyyy
"Transposition ciphers
In a transposition cipher, the letters themselves are kept unchanged, but their order within the message is scrambled according to some well-defined scheme. Many transposition ciphers are done according to a geometric design. A simple (and once again easy to crack) encryption would be to write every word backwards. For example "Hello my name is Alice." would now be "olleH ym eman si ecilA." A scytale is a machine that aids in the transposition of methods.
In a columnar cipher, the original message is arranged in a rectangle, from left to right and top to bottom. Next, a key is chosen and used to assign a number to each column in the rectangle to determine the order of rearrangement. The number corresponding to the letters in the key is determined by their place in the alphabet, i.e. A is 1, B is 2, C is 3, etc. For example, if the key word is CAT and the message is THE SKY IS BLUE, this is how you would arrange your message:
C A T
3 1 20
T H E
S K Y
I S B
L U E
Next, you take the letters in numerical order and that is how you would transpose the message. You take the column under A first, then the column under C, then the column under T, as a result your message "The sky is blue" has become: HKSUTSILEYBE
In the Chinese cipher's method of transposing, the letters of the message are written from right to left, down and up columns to scramble the letters. Then, starting in the first row, the letters are taken in order to get the new ciphertext. For example, if the message needed to be enciphered was THE DOG RAN FAR, the Chinese cipher would look like this:
R R G T
A A O H
F N D E
The cipher text then reads: RRGT AAOH FNDE
Many transposition ciphers are similar to these two examples, usually involving rearranging the letters into rows or columns and then taking them in a systematic way to transpose the letters. Other examples include the Vertical Parallel and the Double Transposition Cipher.
More complex algorithms can be formed by mixing substitution and transposition in a product cipher; modern block ciphers such as DES iterate through several stages of substitution and transposition."
"The Playfair cipher was the first practical digraph substitution cipher. The scheme was invented in 1854 by Charles Wheatstone, but was named after Lord Playfair who promoted the use of the cipher. The technique encrypts pairs of letters (digraphs), instead of single letters as in the simple substitution cipher. The Playfair is significantly harder to break since the frequency analysis used for simple substitution ciphers does not work with it. Frequency analysis can still be undertaken, but on the ~600 possible digraphs rather than the 26 possible monographs. Frequency analysis thus requires much more ciphertext in order to work.
It was used for tactical purposes by British forces in the Second Boer War and in World War I and for the same purpose by the Australians during World War II. This was because Playfair is reasonably fast to use and requires no special equipment. A typical scenario for Playfair use would be to protect important but non-critical secrets during actual combat. By the time the enemy cryptanalysts could break the message the information was useless to them [1].
From Kahn's 'The CodeBreakers':
Perhaps the most famous cipher of 1943 involved the future president of the U.S., J. F. Kennedy, Jr. On 2 August 1943, Australian Coastwatcher Lieutenant Arthur Reginald Evans of the Royal Australian Naval Volunteer Reserve saw a pinpoint of flame on the dark waters of Blackett Strait from his jungle ridge on Kolombangara Island, one of the Solomons. He did not know that the Japanese destroyer Amagiri had rammed and sliced in half an American patrol boat PT-109, under the command of Lieutenant John F. Kennedy, United States Naval Reserve. Evans received the following message at 0930 on the morning of the 2 of August 1943:" (Practicalcryptography.com)
KXJEY UREBE ZWEHE WRYTU HEYFS
KREHE GOYFI WTTTU OLKSY CAJPO
BOTEI ZONTX BYBWT GONEY CUZWR
GDSON SXBOU YWRHE BAAHY USEDQ
The translation:
PT BOAT ONE OWE NINE LOST IN ACTION IN BLACKETT
STRAIT TWO MILES SW MERESU COVE X CREW OF TWELVE
X REQUEST ANY INFORMATION.
I can't really relate this to what we're doing right now, but I'm sure we'll learn this when we start cryptography
It was used for tactical purposes by British forces in the Second Boer War and in World War I and for the same purpose by the Australians during World War II. This was because Playfair is reasonably fast to use and requires no special equipment. A typical scenario for Playfair use would be to protect important but non-critical secrets during actual combat. By the time the enemy cryptanalysts could break the message the information was useless to them [1].
From Kahn's 'The CodeBreakers':
Perhaps the most famous cipher of 1943 involved the future president of the U.S., J. F. Kennedy, Jr. On 2 August 1943, Australian Coastwatcher Lieutenant Arthur Reginald Evans of the Royal Australian Naval Volunteer Reserve saw a pinpoint of flame on the dark waters of Blackett Strait from his jungle ridge on Kolombangara Island, one of the Solomons. He did not know that the Japanese destroyer Amagiri had rammed and sliced in half an American patrol boat PT-109, under the command of Lieutenant John F. Kennedy, United States Naval Reserve. Evans received the following message at 0930 on the morning of the 2 of August 1943:" (Practicalcryptography.com)
KXJEY UREBE ZWEHE WRYTU HEYFS
KREHE GOYFI WTTTU OLKSY CAJPO
BOTEI ZONTX BYBWT GONEY CUZWR
GDSON SXBOU YWRHE BAAHY USEDQ
The translation:
PT BOAT ONE OWE NINE LOST IN ACTION IN BLACKETT
STRAIT TWO MILES SW MERESU COVE X CREW OF TWELVE
X REQUEST ANY INFORMATION.
I can't really relate this to what we're doing right now, but I'm sure we'll learn this when we start cryptography
post for 10/31
What are the different uses of prime factorization? Include some we have done in class and some that you find. Also is it related to cryptography?
Prime factorization can be used for many many things. Some include finding the greatest common factor (GCF), lowest common multiple (LCM), and also divisors of a number.
Using prime factorization and knowing what the uses of it are can help you do all of these things effectively and quickly.
I can also imagine how it would be related to cryptography.
Prime factorization can be used for many many things. Some include finding the greatest common factor (GCF), lowest common multiple (LCM), and also divisors of a number.
Using prime factorization and knowing what the uses of it are can help you do all of these things effectively and quickly.
I can also imagine how it would be related to cryptography.
11/7
"One of the simplest examples of a substitution cipher is the Caesar cipher, which is said to have been used by Julius Caesar to communicate with his army. Caesar is considered to be one of the first persons to have ever employed encryption for the sake of securing messages. Caesar decided that shifting each letter in the message would be his standard algorithm, and so he informed all of his generals of his decision, and was then able to send them secured messages. Using the Caesar Shift (3 to the right), the message,
"RETURN TO ROME"
would be encrypted as,
"UHWXUA WR URPH"
In this example, 'R' is shifted to 'U', 'E' is shifted to 'H', and so on. Now, even if the enemy did intercept the message, it would be useless, since only Caesar's generals could read it.
Thus, the Caesar cipher is a shift cipher since the ciphertext alphabet is derived from the plaintext alphabet by shifting each letter a certain number of spaces. For example, if we use a shift of 19, then we get the following pair of ciphertext and plaintext alphabets:
Plaintext: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Ciphertext: T U V W X Y Z A B C D E F G H I J K L M N O P Q R S
To encipher a message, we perform a simple substitution by looking up each of the message's letters in the top row and writing down the corresponding letter from the bottom row. For example, the message
THE FAULT, DEAR BRUTUS, LIES NOT IN OUR STARS BUT IN OURSELVES.
would be enciphered as
MAX YTNEM, WXTK UKNMNL, EBXL GHM BG HNK LMTKL UNM BG HNKLXEOXL.
Essentially, each letter of the alphabet has been shifted nineteen places ahead in the alphabet, wrapping around the end if necessary. Notice that punctuation and blanks are not enciphered but are copied over as themselves."
The Caesar Cipher is basically just shifting the whole alphabet and doing codes from there on. A could be C. Then C would be E. You get the point.
Source: http://www.cs.trincoll.edu/~crypto/historical/caesar.html
"RETURN TO ROME"
would be encrypted as,
"UHWXUA WR URPH"
In this example, 'R' is shifted to 'U', 'E' is shifted to 'H', and so on. Now, even if the enemy did intercept the message, it would be useless, since only Caesar's generals could read it.
Thus, the Caesar cipher is a shift cipher since the ciphertext alphabet is derived from the plaintext alphabet by shifting each letter a certain number of spaces. For example, if we use a shift of 19, then we get the following pair of ciphertext and plaintext alphabets:
Plaintext: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Ciphertext: T U V W X Y Z A B C D E F G H I J K L M N O P Q R S
To encipher a message, we perform a simple substitution by looking up each of the message's letters in the top row and writing down the corresponding letter from the bottom row. For example, the message
THE FAULT, DEAR BRUTUS, LIES NOT IN OUR STARS BUT IN OURSELVES.
would be enciphered as
MAX YTNEM, WXTK UKNMNL, EBXL GHM BG HNK LMTKL UNM BG HNKLXEOXL.
Essentially, each letter of the alphabet has been shifted nineteen places ahead in the alphabet, wrapping around the end if necessary. Notice that punctuation and blanks are not enciphered but are copied over as themselves."
The Caesar Cipher is basically just shifting the whole alphabet and doing codes from there on. A could be C. Then C would be E. You get the point.
Source: http://www.cs.trincoll.edu/~crypto/historical/caesar.html
week 4 prompt
They have a man by the name of Petros Petrosyan who is a famous cipher and who has solved the great pyrmid cipher that remained unsolved for so many years. This is a cipher code on how to pyramids are so closely connected with the way of problem solving today and how they connect with the modern day math of ancient egypt. It has a great amount of geometry within the codes and how to slove them. I think that the reason for this is because it was something that was built and geometry is the foundation of all the building supplies needed in life. I dont think that it has related to anything that we have learned this year but I think it does have something to do with what we learned in Geometry.
www.hermann-uwe.de/blog/famous-unsolved-codes-and-ciphers
www.hermann-uwe.de/blog/famous-unsolved-codes-and-ciphers
Alaina's blog, 7 Nov. 2010
"Steganography is the art of hiding the existence of a message rather than its meaning. It is currently a very HOT topic as one subdivision of steganography is digital watermarking, a technique used to copyright and to protect digital images, music and software."
Older methods of steganography include writing in lemon juice or tracing words through a piece of paper so that they have to be colored or shaded over to be seen.
Stealthy, right?
SOURCE:
http://home.cogeco.ca/~cipher/
Older methods of steganography include writing in lemon juice or tracing words through a piece of paper so that they have to be colored or shaded over to be seen.
Stealthy, right?
SOURCE:
http://home.cogeco.ca/~cipher/
Stephen's Post 11/7/2010
"In cryptography, a substitution cipher is a method of encryption by which units of plaintext are replaced with ciphertext according to a regular system; the "units" may be single letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth. The receiver deciphers the text by performing an inverse substitution.
Substitution ciphers can be compared with transposition ciphers. In a transposition cipher, the units of the plaintext are rearranged in a different and usually quite complex order, but the units themselves are left unchanged. By contrast, in a substitution cipher, the units of the plaintext are retained in the same sequence in the ciphertext, but the units themselves are altered."
It seems that this is one of the most famous methods of ciphering, or deciphering, to say the least.
"There are a number of different types of substitution cipher. If the cipher operates on single letters, it is termed a simple substitution cipher; a cipher that operates on larger groups of letters is termed polygraphic. A monoalphabetic cipher uses fixed substitution over the entire message, whereas a polyalphabetic cipher uses a number of substitutions at different times in the message, where a unit from the plaintext is mapped to one of several possibilities in the ciphertext and vice-versa."
Source: http://en.wikipedia.org/wiki/Substitution_cipher
Chase 11/7
The cipher disk was invented in 1467 by Leon Battista Alberti, a famous Italian philosopher and architect.
Alberti used two different alphabets located on concentric rings - this means one ring is inside of or on top of another. By lining up two different letters, one from each ring, he could make a simple substitution alphabet in which he could create a cipher.
For example, if he aligned the A on the outer ring with the G on the inner ring, this would make the following substitution alphabet used to encrypt a message:
OUTER RING: ABCDEFGHIJKLMNOPQRSTUVWXYZ
INNER RING: GHIJKLMNOPQRSTUVWXYZABCDEF
From there, he could encrypt his message and send it to someone who knew the secret to revealing the message.
http://www.nsa.gov/kids/ciphers/ciphe00005.shtml
Alberti used two different alphabets located on concentric rings - this means one ring is inside of or on top of another. By lining up two different letters, one from each ring, he could make a simple substitution alphabet in which he could create a cipher.
For example, if he aligned the A on the outer ring with the G on the inner ring, this would make the following substitution alphabet used to encrypt a message:
OUTER RING: ABCDEFGHIJKLMNOPQRSTUVWXYZ
INNER RING: GHIJKLMNOPQRSTUVWXYZABCDEF
From there, he could encrypt his message and send it to someone who knew the secret to revealing the message.
http://www.nsa.gov/kids/ciphers/ciphe00005.shtml
kaitlynnn's blogg
In 1885, a pamplet was published supposedly telling where to find a bunch of gold and silver. The instructions were encrypted. The pamplet told the story of Thomas Jefferson Beale, who made a fortune out west, and buried it in Bedford County, Virginia. Three separate messages were encrypted in the pamplet, and consist of pages of numbers, separated by commas. He gave the solution to the second message.
The key to decrypting the second message was the Declaration of Independence. Every number in the message refers to the first letter of a word in it. The third message is relatively uninteresting to treasure hunters, as it tells who the treasure belongs to. Apparently, no one has ever decrypted the first or third message.
There are a few good reasons to think that the pamplet is a hoax. For example, the first message would seem to use the same message of encryption as the second, but with a few much larger numbers, implying that the key document is much longer than the Declaration of Independence. But no matter how long a text is, you probably would not have to search 2000 words deep to find any letter you wanted. Besides, there is internal evidence that the key is actually the Declaration of Independence, and that the original message is mostly meaningless letters with no words, and with a few large numbers
The key to decrypting the second message was the Declaration of Independence. Every number in the message refers to the first letter of a word in it. The third message is relatively uninteresting to treasure hunters, as it tells who the treasure belongs to. Apparently, no one has ever decrypted the first or third message.
There are a few good reasons to think that the pamplet is a hoax. For example, the first message would seem to use the same message of encryption as the second, but with a few much larger numbers, implying that the key document is much longer than the Declaration of Independence. But no matter how long a text is, you probably would not have to search 2000 words deep to find any letter you wanted. Besides, there is internal evidence that the key is actually the Declaration of Independence, and that the original message is mostly meaningless letters with no words, and with a few large numbers
Mal's Post
"Probably the most well known cipher of all time is the German Enigma of World War II. German military and diplomatic information was encoded using an typewriter style machine called “Enigma.” This device had three wheels which would type a different letter than the one you choose on the keyboard. To further complicate the encryption, after each character, the wheels would change so that a different letter would be produced for repeated characters. This made the classic style of decryption and code breaking impossible to complete on an Enigma created document."
So. my understanding of Enigma is this. to decrypt a code, the receiver has to know which wheel to use and where to start on the wheel. Apparently this created a barrier during the war for the enemies. England deciphered the code, but kept it secret until like 30 years ago.
I don't know exactly how this is related to what we've been doing in number theory. Probably something to do with the probability that it would encrypt a letter for another letter given several wheels.
So. my understanding of Enigma is this. to decrypt a code, the receiver has to know which wheel to use and where to start on the wheel. Apparently this created a barrier during the war for the enemies. England deciphered the code, but kept it secret until like 30 years ago.
I don't know exactly how this is related to what we've been doing in number theory. Probably something to do with the probability that it would encrypt a letter for another letter given several wheels.
Feroz's Blog
The Advance Encryption Standard, also known as Rijndael.
Source:
http://en.wikipedia.org/wiki/Advanced_Encryption_Standard
"Rijndael is the block cipher algorithm recently chosen by the National Institute of Science and Technology (NIST) as the Advanced Encryption Standard (AES). It supercedes the Data Encryption Standard (DES). NIST selected Rijndael as the standard symmetric key encryption algorithm to be used to encrypt sensitive (unclassified) American federal information. The choice was based on a careful and comprehensive analysis of the security and efficiency characteristics of Rijndael's algorithm."
From what I understand it's used by the government, and it's made by two Belgian dudes.
Source:
http://en.wikipedia.org/wiki/Advanced_Encryption_Standard
11/7
Voynich Manuscript - At least 400 years old, this is a 232-page illuminated manuscript entirely written in a secret script. It is filled with drawings of unidentified plants, herbal recipes of some sort, astrological diagrams, and many small human figures in strange plumbing-like contraptions. The script is unlike anything else in existence, but is written in a confident style, seemingly by someone who was very comfortable with it. In 2004 there were some compelling arguments which described a technique that would seemingly prove that the manuscript was a fake, but to date, none of the described techniques have been able to replicate a single section of the Manuscript, so speculations continue.
source:
http://elonka.com/UnsolvedCodes.html
source:
http://elonka.com/UnsolvedCodes.html
11/7/10
Okay, I'm going to talk about The Famous Beale Cipher. It states that in the year 1885 some sort of pamphlet was publish telling where to find gold and silver that's worth millions of dollars. Here's the problem, though. All the instructions are encrypted (in code). Anyway, it is called the Famous Beale Cipher on behalf of Thomas Jefferson Beale who somehow made a large fortune and buried it Bedford County, Virginia. In the pamphlet, there were three separate messages..telling exactly where the gold and silver was, how much of it was there, who it originally belonged to, etc..Actually the writer of the pamplet solved the second message. You see, the messages are written entirely in numbers with commas between them. He said the way that it was arranged you had to use the Declaration of Independence. As in, whatever number was there, that number represented the first letter of a word in the Declaration of Independence. For example, if you see the number 7, that means that you find the 7th word of the Declaration of Independence and take the first letter of that word. Of course after you find all the letters, you have to somehow figure out what the words are by trying different things..But, to this day, the puzzle still has not been solved.
Source:
Loy, Jim. 2001. The Famous Beale Cipher. 7 Nov. 2010. http://www.jimloy.com/puzz/beale.htm.
Source:
Loy, Jim. 2001. The Famous Beale Cipher. 7 Nov. 2010. http://www.jimloy.com/puzz/beale.htm.
Saturday, November 6, 2010
Week 4 Blog Prompt
Research a famous cipher method online. Everyone needs to have different sources and types. Briefly explain the method and how it works to the best of your understanding. Does it use anything similar to what we have done so far this year?
Friday, November 5, 2010
Number Theory Blogs
The reason I didn't do a blog for the weekend of 10/30 - 10/31 is because I went on the blog on Saturday, and they had no prompt. So, I figured that there wasn't a blog just like all the other weeks...
Thursday, November 4, 2010
Alaina's blog, 31 october 2010
Prime factorization can be used to find many things. It's about breaking things down, going back to the basics.It's used for things like finding lcm, gcd, exponential form, number of divisors, product of positive divisors and to simplify fractions, etc.Cryptography is used to break down & decipher codes. prime factorization is used to break down numbers. Sounds pretty similar. Well I’m sure, in some strange way, it is used in cryptography.
Sunday, October 31, 2010
Mal'ssssss
Okay. well. Prime factorization can be used to find a lot of things, I'm assuming. However, the primary purposes in our class is just finding the least common multiple (LCM) and the greatest common divisor (GCD). FOR EXAMPLE, if you were given two numbers and told to find the LCM, you would just find the prime factorization and multiply every number (highest degree between both). Easy enough right? Well, GCD is just multiply all the numbers they have in common to the lowest degree. Also, prime factorization can be used in practical situations when like trying to find how many prime numbers of marbles can I put in a cup...etc.
This prime factorization is related to crytography in that, most likely, different variations of prime numbers/factorization will be used when categorizing each letter.
OKAY..so I'm done. kthnxbye.
This prime factorization is related to crytography in that, most likely, different variations of prime numbers/factorization will be used when categorizing each letter.
OKAY..so I'm done. kthnxbye.
ahhh
Prime factorization can be used to find number of diviors, what the divisors are, divisors that are multiples of 2 or square root of 3 or perfect squares and all that good stuff. you can use it to simplify fractions by breaking down the number on top and the one on bottom.
question 2 .
okay, we are taking number theory/cryptography right? and what are we learing in this class i speak of, oh, that's right, prime factorization, so they obviously have something to do with each other, haha.
and do these blogs have to be 150 words too? becuase i remeber on previous blog prompts that they didn't require long answers.
question 2 .
okay, we are taking number theory/cryptography right? and what are we learing in this class i speak of, oh, that's right, prime factorization, so they obviously have something to do with each other, haha.
and do these blogs have to be 150 words too? becuase i remeber on previous blog prompts that they didn't require long answers.
Halloween Blog emohgniogm'i,syuguoywercs#
What are the different uses of prime factorization?
Uhh.......finding LCM's and GCD's?
Is it realted to cryptography?
possibly.......if I knew what cryptography was, I'd give you a definite answer though...
Uhh.......finding LCM's and GCD's?
Is it realted to cryptography?
possibly.......if I knew what cryptography was, I'd give you a definite answer though...
halloween post.
Prime factorization can be used to find many things. it's about breaking things down.. going back to the basics.
it's used for things like finding lcm, gcd, ... exponential form, # of divisors, product of divisors... & to simplify fractions, etc.
cryptography is used to break down & decipher codes. prime factorization is used to break down numbers.... sounds similar right? well i'm sure, in some strange way, it is used in cryptography. we'll probably learn how to do that in the 2nd semester.
it's used for things like finding lcm, gcd, ... exponential form, # of divisors, product of divisors... & to simplify fractions, etc.
cryptography is used to break down & decipher codes. prime factorization is used to break down numbers.... sounds similar right? well i'm sure, in some strange way, it is used in cryptography. we'll probably learn how to do that in the 2nd semester.
Helen's Bloggggg
What are the different uses of prime factorization? Is it realted to cryptography?
In class we use prime factorization to find the gcd, lcm, number of divisors, the product of the divisors, and many others things that I don't know how to do :). I think it is related to cryptography because it might help codes and stuff and it makes the number simpler.
In class we use prime factorization to find the gcd, lcm, number of divisors, the product of the divisors, and many others things that I don't know how to do :). I think it is related to cryptography because it might help codes and stuff and it makes the number simpler.
Chapter 10
This past week, we have started off learning the formulas needed in Chapter 10.In order to find the right answers, you MUST know your trig chart since there is no calculators being used. You also have to know your formulas. You need these when finding the exact value, simplifying the expression, etc.
List:
List:
- Cos(A+/-B)=CosACosB-/+SinASinB
- Sin(A+/-B)=SinASinB+/-CosBCosA
- Tan(A+B)=TanA+TanB/1-TaNATanB
- Tan(A-B)=TanA-TanB/1+TanATanB
Double Angles:
- Sin2A=2SinAcosA
- Cos2A=Cos2A-Sin2A
- Cos2A=Cos2A-1
- Cos2a=1-Sin2A
- Tan2A=2TanA/1-Tan2A
Half Angles:
- Sin2a=+/-(Square root)1-cosA/2
- cos2a=+/-(sqaure root)1+cosA/2
- tan2a=1-cosA/sinA=sinA/1+cosA
We use these formulas to find out these unknown trig functions ex:sin75degrees
Prime Factorizatioin
In Number Theory, We use the prime factorization method for many parts of math.
- Finding the GCM and LCM of sets and numbers
- Finding how many divisors a number has
- To find the product of the divisors of numbers using n^t(n)/2
- Place numbers in expo form
- Find the prime fact.
We used prime factorization because it would be helpful in breaking down large numbers into certain groups easier than some other methods.
I'm certain that Prime Factorization is related in cryptology since cryptology has something to do with breaking and deciphering codes using conventional methods. Prime Factorization is used in other maths to help solve numbers easily in different problems
Halloween Blog of the Dead -topsgolbredrum
What are the different uses of the uses of the uses of prime factorizationings? Oh I'll tell ya. I'll tell ya alright. We use prime factoriaztions for many a thing in B-Rob's 5th hour numbr theory class. We use it simply to use it. We use it simply not to use it. We use it to find the greatest cmmon denominators and least common mulltiples of 2 or more dieffernt numbers. It is a eraly col method of findngig numbers and ish tha is usefull. we also use prime ffactoriazaitonns to find the total number of divisors , perfect swuares and cubes and products of the greatest commmon denominators and least common mulltiples of two or more numbers. you can also simplify fractionss by using the method of prinme factorizations.
Is it realted to cryptographY?
yea i think it is but not in a very applicable way, ya know what im sayin. maybe you gota break down numbes and stuff to find a diffetrnt way to deciperh a code or somehting lkike that. You gotta break down the numbers in a way to try to simplify them into somehthing simpler. that's all i really got right now i can't think of anything else.
10/31/10 CHASE :D
what are the different uses of prime factorization? some uses are for finding LCM, GCD, the number of divisors in a number, the primes a number is divisible by, finding the product of divisors of a number, and putting a number in exponential notation.
is prime factorization used in cryptography? im sure it is. cryptography... as far as i know, is the breakdown of hidden codes through numbers; im sure prime factorization plays its role in some forms of cryptography because it simplifies much more complicated numbers down to its primes, and that could be VERY useful in cracking code
is prime factorization used in cryptography? im sure it is. cryptography... as far as i know, is the breakdown of hidden codes through numbers; im sure prime factorization plays its role in some forms of cryptography because it simplifies much more complicated numbers down to its primes, and that could be VERY useful in cracking code
Kaitlyn's blogggg
What are the different uses of prime factorization? Well in class, we have been using prime factorization for things like finding the gcd, lcm, numbers of divisors, and other very complicated things. Prime factorization make it easier to figure out different problems because it simplifys a number down. I'm pretty sure we will be using prime factorization in crytography. We might use it to help us decode different things because prime factorization simplifys the number down, this would make it easier to do. I'm sure that prime factorization is used in many different types of math and we will be using it all the time now.(:
10/31/10
*Some uses for prime factorization (that we've done in class) are as follows:
-to find the GCD/LCM of a set of numbers
-to find what prime numbers a number is divisible by
-to find the number of divisors a number has
-to find the product of the divisors of a number using the formula n^t(n)/2
-to put a number in exponential notation
-to basically break down very large numbers into work that is easier to decipher
*Is prime factorization related to cryptography?
Yes it sure is. Cryptography is basically the study of secret codes; how to make and break them..As far as I know anyway. And in order to break them, I'm sure you have to know the prime factorization of numbers to make it simpler and easier to understand. Of course if you were dealing with very large numbers, it would take a long time to decode them (or whatever you will have to do..). Using prime factorization just seems like it would make the process quicker and simpler. And I'm sure in cryptography there are many other methods to use besides the prime factorization, such as for instance, Bezout's Theorem and the Euclidean Algorithm...(And no I didn't already know this, I'm not that good, haha. I kinda sorta got this off of our topic list for the class :P)
-to find the GCD/LCM of a set of numbers
-to find what prime numbers a number is divisible by
-to find the number of divisors a number has
-to find the product of the divisors of a number using the formula n^t(n)/2
-to put a number in exponential notation
-to basically break down very large numbers into work that is easier to decipher
*Is prime factorization related to cryptography?
Yes it sure is. Cryptography is basically the study of secret codes; how to make and break them..As far as I know anyway. And in order to break them, I'm sure you have to know the prime factorization of numbers to make it simpler and easier to understand. Of course if you were dealing with very large numbers, it would take a long time to decode them (or whatever you will have to do..). Using prime factorization just seems like it would make the process quicker and simpler. And I'm sure in cryptography there are many other methods to use besides the prime factorization, such as for instance, Bezout's Theorem and the Euclidean Algorithm...(And no I didn't already know this, I'm not that good, haha. I kinda sorta got this off of our topic list for the class :P)
Justin
We use prime factorization for things like GCD, LCM, number of divisors, product of divisors. It makes problems easier because youre shrinking it down into powers of primes. I think we are going to use prime factorization for cryptography, but I don'r really know how.
There are multiple uses of prime factorization, though we have only gone over a few. One way we used prime factorization was to find the least common multiple, greatest common denominator, and finding how many divisors and so forth.
Finding the prime factorization is very helpful becasue it is often easier to work with smaller numbers rather than doing intense math.
I believe that cryptogrophy will use prime factorizations to help decode the problems because you may be able to find a code and be able to realize that the gcd equals one letter than the same for antoehr.
It is very important that we learn to do the prime factorization of numbers because it is used in a varitety of problems.
Finding the prime factorization is very helpful becasue it is often easier to work with smaller numbers rather than doing intense math.
I believe that cryptogrophy will use prime factorizations to help decode the problems because you may be able to find a code and be able to realize that the gcd equals one letter than the same for antoehr.
It is very important that we learn to do the prime factorization of numbers because it is used in a varitety of problems.
Blog Prompt
Jordan's computer is currently down. Thus I will be posting this week's blog prompt.
What are the different uses of prime factorization? Include some we have done in class and some that you find. Also is it related to cryptography?
What are the different uses of prime factorization? Include some we have done in class and some that you find. Also is it related to cryptography?
Wednesday, October 20, 2010
Feroz's Blog
I feel like I've mastered quite a bit in this class, like the sieve, finding the gcd and lcm, and definitely prime factorization.
I struggled with the word problems, like "find the number the divisors of 100,000,000,000 that are perfect squares" and stuff like that. Just not to that extent.
I never really took this class seriously, and I still don't, so I should probably do homework and rework problems to improve my grade.
I struggled with the word problems, like "find the number the divisors of 100,000,000,000 that are perfect squares" and stuff like that. Just not to that extent.
I never really took this class seriously, and I still don't, so I should probably do homework and rework problems to improve my grade.
Sunday, October 17, 2010
DaViD
I've mastered alot in this class. The seive is easy, just takes alot of work and writing numbers. prime factorization is something that is now super easy to understand and work. I can work matrices and logs. I struggled with some of the weird word problems we had in homework, and one way i can fix my grade and fix that is possibly doing that homework more often and to the fullest extent. I will be doing amazing in this class from now on because i want the food prize EVERYTIME we cipher. amen.
Blog for 10/17
What concept do you feel like you mastered this nine weeks?
Hmm, I honestly don't think I've "mastered" any concepts this nine weeks, but I've started to get better with the whole LCM AND GCD thing
What concept did you struggle with the most? Why?
well, I have no clue actually, I'd have to go with finding prime numbers because I'd always forget one little common mistake and get the numbers wrong
What can you change this nine weeks in your study habits, etc to improve your grade?
I'd have to go with doing the homework twice I guess
Hmm, I honestly don't think I've "mastered" any concepts this nine weeks, but I've started to get better with the whole LCM AND GCD thing
What concept did you struggle with the most? Why?
well, I have no clue actually, I'd have to go with finding prime numbers because I'd always forget one little common mistake and get the numbers wrong
What can you change this nine weeks in your study habits, etc to improve your grade?
I'd have to go with doing the homework twice I guess
Muuraaayyy
The concept I mastered is prime factorization. It was easy to begin with, but now I can go really fast and it doesn't take near as long. I get alot of the processes we learned, and I will most likely remember them because they make sense, it easy( well not easy but yea) to figure it out with some thinking. I struggled alot with logs! I will never get them! What i could do to help is perhaps go to alpha practice with shawn... and at first, i had heard that the math extra class was just to focus math club, not this. So i'm ready now. I know my study habits wil change, in fact, they could get reall, because i already focus all homework time on advanced math, so it that class gets harder, i'll really be in trouble, with that one and this one. anyway, this nine weeks was rough, but at least i can find the all the prime factors between 1 and 100. haha,
MalPal
Okay. so the following:
What concept do you feel like you mastered this nine weeks?
Umm..Hello? Prime factorization. It gets intense sometimes, but you don't even have to really think about it, or at least I don't. I just divide and divide and divide till no end. So, to answer the question, I have indeed mastered prime factorization.
What concept did you struggle with the most? Why?
I struggled with the whole let's find all the multiples of whatever under 10000 that are perfect squares. I think I don't get this because I never know like what to put the prime factorization over like what you do with
What can you change this nine weeks in your study habits, etc to improve your grade?
Rework problems
What concept do you feel like you mastered this nine weeks?
Umm..Hello? Prime factorization. It gets intense sometimes, but you don't even have to really think about it, or at least I don't. I just divide and divide and divide till no end. So, to answer the question, I have indeed mastered prime factorization.
What concept did you struggle with the most? Why?
I struggled with the whole let's find all the multiples of whatever under 10000 that are perfect squares. I think I don't get this because I never know like what to put the prime factorization over like what you do with
What can you change this nine weeks in your study habits, etc to improve your grade?
Rework problems
Justin 6
What did i master this 9 weeks? Hehe prime factorization related stuff which includes gcd lcm #of divisors product of divisors and so on. i like this stuff cause i grasp it quickly and work relatively fast with it. what i could do to improve my grade would be to practice more so i'll be more accurate and quick. other than that, i think i did pretty good this 9 weeks
kmart(:
What i feel i've mastered the most this nine weeks is lcm and gcd. This is reallyy easy because all you have to do is find the prime factorization of your number and solve for it. It takes me no time at all to do this, i could probably do it in my sleep. Also with finding the prime numbers between a set of numbers like 1-100. The concept i struggled the most with was when we got more into the prime factorization concept, like finding the divisors and other things like that. It's really confusing for me and i'm not really good at understanding exponents. Other than that, i think i did pretty well this nine weeks.
Helen Melon :)
What I think that I mastered this nine weeks is how to find the gcd and lcm of two or more numbers by using the prime factorizations of the number. I also learned how to find the prime number between 1-100 by using the sieve. What I struggle the most with is finding the multiple divisors, conics, and exponents. What I should do to improve my grades is to pay more attention and study more
Alaina's blog, 17 October 2010
What concept do you feel like you mastered this nine weeks? What concept did you struggle with the most? Why? What can you change this nine weeks in your study habits, etc to improve your grade?
The concepts that we learned this nine weeks reflect on childhood math (elementary math). The GCD and LCM we learned in probably the 4th grade and had to learn again in 10th, 11th, and 12th grades. The concepts are fairly simple, but because we have been doing things much harder for a lot of years, we make them more complicated than they are.
The concept that I struggled with this nine weeks would be the product of all positive divisors of some number n. I can never remember the formula for this; therefore, I spend the majority of my time on tests doing this the long way. And I finally give up on it and move on to simpler problems.
This nine weeks, I could have studied a little bit more for the tests and quizzes in this class. Studying more would have improved my grades. I also procrastinate tremendously! So I need to work on these things.
The concepts that we learned this nine weeks reflect on childhood math (elementary math). The GCD and LCM we learned in probably the 4th grade and had to learn again in 10th, 11th, and 12th grades. The concepts are fairly simple, but because we have been doing things much harder for a lot of years, we make them more complicated than they are.
The concept that I struggled with this nine weeks would be the product of all positive divisors of some number n. I can never remember the formula for this; therefore, I spend the majority of my time on tests doing this the long way. And I finally give up on it and move on to simpler problems.
This nine weeks, I could have studied a little bit more for the tests and quizzes in this class. Studying more would have improved my grades. I also procrastinate tremendously! So I need to work on these things.
10/17/2010
What concept do you feel like you mastered this nine weeks? What concept did you struggle with the most? Why? What can you change this nine weeks in your study habits, etc to improve your grade?
This nine weeks I feel that I have mastered many new concepts that I never thought I would learn or have to learn. The concepts are basically elementary concepts, and I never knew something so complicated could be so simple, and I never knew something so simple could be so complicated.
Anyway, I feel that I definitely mastered the concept of finding prime factorizations for different numbers. Also, I feel totally comfortable with LCM and GCD now, more than ever before in my life, I feel like a pro at it.
The concept I struggled with the most this nine weeks would probably have to be conics, exponents, and finding the exact factors of a number once you find the prime factorization. I've always had trouble with conics, especially parabolas for some reason, exponents are ok, but sometimes they can be complicated, and finding those factors just takes a long time and I get lazy and don't feel like doing them I guess.
What I can change the next nine weeks is definitely just study more and more every night, and stop procrastinating in class and actually try to pay attention to the subject material being taught.
10/17/10
The concept I feel like I've mastered this nine weeks is definitely prime factorization, and using the prime factorization of numbers to find the lcm and gcd. That is extremely easy for me now and I don't even have to think about it. I know exactly what I have to do by simply glancing at the problem. I also feel I've gotten better with recognizing what you have to find in word problems. (For instance, if they're asking for the factors of a number, the gcd, etc.) The concept I struggled with the most I think is in Chapter 5..It has to with the number of divisors a number has, but that's all they give you in a problem and they ask you to find, for example, x^3 if x has 14 divisors. Problems like that still confuse me and I could use more practice with them...To improve my grade I can study more thoroughly the night before and actually rework the problems from the review pages, to make sure I know what I'm doing. Other than that, I think I'm doing okay in the class..it just takes me a little while to catch on to things ha
Recap Post
This nine weeks i felt very comfortable with most of the topics. I especially felt comfortable with the factor trees and GCDs and LCMs. The one thing i was not comfortable with was when to do the t^n/2 formula. That is definately the easiest thing ever and most people are probably laughing at me, but honestly, i never knew when to use it. Another thing i felt comfortable with was the groups of pennies and then how many different ways..that stuff was easy.
This nine weeks went by pretty fast with some pretty easy terms and topics. Hopefully the next three are the same!
This nine weeks went by pretty fast with some pretty easy terms and topics. Hopefully the next three are the same!
Post
I feel like I have mastered how to use the prime factorization to find the gcd and the lcm of two or more numbers. This is a much easier process then trying to compare all of numbers. I still am having problems with when they tell you to find the multiples of some number between two numbers using the prime factorization, so instead I just use the sieve. I think that I can improve my grades by looking over what we did in class and doing my homework every night.
ReCaP
The lesson in number theory that i have i mastered the most is finding the greatest common divisor(GCM) and the least common multiple(LCM) of numbers by using prime factorization. Before NUmber theory, i knew how to find them but it took me alot longer than it should have. I stuggle to find the product of the LCM of a number. I tend to get that confused with finding common divisors. Maybe i should stop kidding around in class and ask more relevant questions pertaining to the lesson.
Saturday, October 16, 2010
postttt.
What I feel like i've mastered this nine weeks is how to find lcm and gcd. i always knew how, but i would get it confused between the two of them. from so much practice, i don't think i will ever forget it now. I struggle with how to find gcd if given the lcm. I don't understand how that's possible. I need to take more legible notes, and maybe I could understand what i was actually writing down.
Thursday, October 14, 2010
4 most important numbers
Hmm For my four most important numbers I would have to go with pi, and e. As far as two others I would go with the golden ratio because of the significance that it holds and i (which is imaginary but still pretty important... I'm not sure if I can count i..) Pi because almost all geometry and trig need an approximation of pi so it is used in tons of applications, and e for a similar reason.
Just thought I would add the thoughts off the top of my head to this one...
Just thought I would add the thoughts off the top of my head to this one...
Blog Prompt
What concept do you feel like you mastered this nine weeks? What concept did you struggle with the most? Why? What can you change this nine weeks in your study habits, etc to improve your grade?
Monday, October 11, 2010
Some comments
I just wanted to note that for the people that got 1/4 for the gcd*lcm problem, you were "correct" in method but incorrect in answer. I meant for the problem to read that a*b = 4096, so it was my fault, but the real problem with the whole thing is that no set of numbers exist satisfying the criteria I gave. I meant for the answer to be 4. Sorry about that.
I'll also comment that I feel like a lot of you (and by a lot of you I mean EVERY LAST ONE OF YOU) were VERY close-minded and didn't give any thought to the most important numbers in mathematics at ALL.
I assure you that there are FAR more important numbers out there than 4, 5, 100, etc. You guys REALLY should strive to think outside of the box sometimes. It won't hurt you and you might learn a thing or two really.
I'll also comment that I feel like a lot of you (and by a lot of you I mean EVERY LAST ONE OF YOU) were VERY close-minded and didn't give any thought to the most important numbers in mathematics at ALL.
I assure you that there are FAR more important numbers out there than 4, 5, 100, etc. You guys REALLY should strive to think outside of the box sometimes. It won't hurt you and you might learn a thing or two really.
Sunday, October 10, 2010
alaina's blog, 10 oct. 2010
Set 1 abstract
question 1: what is the relationship between the gcd and the lcm of two positive integers, a and b?
-the gcd and lcm have common multiples and, usually, the lcm and be divided by the gcd.
for example: (40, 24)
the prime factorization of 40= 2^3(5)
the prime factorization of 24= 2^3(3)
the lcm of 40 and 24 is 120
the gcd of 40 and 24 is 8
question 2: if you were asked to state what you think to be the four most important numbers of all mathematics. i'm not asking for your luck number, or 42, or 1337, or any other quirky/"clever" response. i want to know, mathematically speaking, what you think the four most important numbers are. justify your answer.
-i really do think the four most important numbers are 1,2,3, and 4 because ,first, these are the first numbers one learns as a child. second, when added in any combination, these numbers can form all other numbers 5-10. all numbers 1-10 are the basis for all other numbers, 11-infinity.
Set 2 combination:
question 1: what is the gcd and lcm of the numbers 36 and 84? what is the product of 36 and 84?
-3squared(2squared)=36; 2squared(3)(7)=84.
the gcd of 36 and 84 is 12
the lcm of 36 and 84 is 252
the product of 36 and 84 is 3024
question 2: if the product of two numbers, a and b, is 1024 and the lcm is 4096, what must their gcd be?
their greatest common divisor is 1024.
question 1: what is the relationship between the gcd and the lcm of two positive integers, a and b?
-the gcd and lcm have common multiples and, usually, the lcm and be divided by the gcd.
for example: (40, 24)
the prime factorization of 40= 2^3(5)
the prime factorization of 24= 2^3(3)
the lcm of 40 and 24 is 120
the gcd of 40 and 24 is 8
question 2: if you were asked to state what you think to be the four most important numbers of all mathematics. i'm not asking for your luck number, or 42, or 1337, or any other quirky/"clever" response. i want to know, mathematically speaking, what you think the four most important numbers are. justify your answer.
-i really do think the four most important numbers are 1,2,3, and 4 because ,first, these are the first numbers one learns as a child. second, when added in any combination, these numbers can form all other numbers 5-10. all numbers 1-10 are the basis for all other numbers, 11-infinity.
Set 2 combination:
question 1: what is the gcd and lcm of the numbers 36 and 84? what is the product of 36 and 84?
-3squared(2squared)=36; 2squared(3)(7)=84.
the gcd of 36 and 84 is 12
the lcm of 36 and 84 is 252
the product of 36 and 84 is 3024
question 2: if the product of two numbers, a and b, is 1024 and the lcm is 4096, what must their gcd be?
their greatest common divisor is 1024.
10/10/10 Post (Binary FTW)
Set 1, abstract:
Question 1.) What is the relationship between the gcd and lcm of two positive integers, a and b?
hmmm, I honestly have no clue at all
Question 2.) If you were asked to state what you think to be the four most important numbers to all of mathematics. I'm not asking for your lucky number, or 42, or 1337, or any other quirky/"clever" response. I want to know, mathematically speaking, what you think the four most important numbers are. P.S., the answer is not 1,2,3, and 4. (Also, there are no right or wrong answers, and if you REALLY think it's 1,2,3,4, just justify your answer VERY well.)
Justify briefly.
well, I'd have to say 0, 10, 100, and 1000 are probably the most important numbers in my opinion
just think about it, multiply anything times 0 and you get 0, or add anything to 0 and the sum doesn't change from the original number
as for 10, 100, and 1000, they create the easiest numbers to multiply and divide by, simple as that, as well as in addition and subtraction
Set 2, concrete:
Question 1.) What is the lcm and gcd of the numbers 36 and 84? What is the product of 36 and 84?
lcm: 252 gcd: 12 3024
Question 2.) If the product of two numbers, a and b, is 1024 and their lcm 4096, what must their gcd be?
4?
Question 1.) What is the relationship between the gcd and lcm of two positive integers, a and b?
hmmm, I honestly have no clue at all
Question 2.) If you were asked to state what you think to be the four most important numbers to all of mathematics. I'm not asking for your lucky number, or 42, or 1337, or any other quirky/"clever" response. I want to know, mathematically speaking, what you think the four most important numbers are. P.S., the answer is not 1,2,3, and 4. (Also, there are no right or wrong answers, and if you REALLY think it's 1,2,3,4, just justify your answer VERY well.)
Justify briefly.
well, I'd have to say 0, 10, 100, and 1000 are probably the most important numbers in my opinion
just think about it, multiply anything times 0 and you get 0, or add anything to 0 and the sum doesn't change from the original number
as for 10, 100, and 1000, they create the easiest numbers to multiply and divide by, simple as that, as well as in addition and subtraction
Set 2, concrete:
Question 1.) What is the lcm and gcd of the numbers 36 and 84? What is the product of 36 and 84?
lcm: 252 gcd: 12 3024
Question 2.) If the product of two numbers, a and b, is 1024 and their lcm 4096, what must their gcd be?
4?
Sunday the 10th
Well, since i read that post and i am pretty tired at the moment, I'm gonna not do this. So, I'm gonna use my homework pass.*uses homework pass*
10/10/10 omg triples
abstract:
1. I think the relationship of the GCD and LCM of a and b is that they are both multiples of a and b
2.In my opinion the 4 most important numbers in math could be 2,3,5,7 because we use prime factorization a whole lot in number theory, which applies to several types of math and these are the 4 smallest primes used for prime factorization
concrete:
1.what is the lcm and gcd of 36 and 84? what is the product?
36
2^2x3^2
84
2^2x3x7
lcm=2^2x3^2x7=252
gcd=12
36x84=3024
lcmXgcd=3024
2.(a)(b)=1024
lcm=4096
well given the information about 36 and 84 i come up with 1/4 sooooooooo... hah
1. I think the relationship of the GCD and LCM of a and b is that they are both multiples of a and b
2.In my opinion the 4 most important numbers in math could be 2,3,5,7 because we use prime factorization a whole lot in number theory, which applies to several types of math and these are the 4 smallest primes used for prime factorization
concrete:
1.what is the lcm and gcd of 36 and 84? what is the product?
36
2^2x3^2
84
2^2x3x7
lcm=2^2x3^2x7=252
gcd=12
36x84=3024
lcmXgcd=3024
2.(a)(b)=1024
lcm=4096
well given the information about 36 and 84 i come up with 1/4 sooooooooo... hah
Kaitlyn's Post 10/10/10
Set 1 abstract:
Question 1) They both find common numbers that are related to both a and b. The lcm can be divided by both a and b. The gcd can go into both a and b.
Question 2) I think the four most important numbers are 0, 1, 2, and 5. 0 because anytime you divide by 0 or multiply by 0, it will end up being no solution or 0. 1 is important because 1 can go into any number at all. When you divide a number by 1 it ends up being that number. Same when you divide by 1. 2 is important because half the numbers in the world can be divided by 2, since 2 goes into all even numbers. 5 is important because when you are dealing with prime factorization, this number shows up a lot.
Set 2 concrete:
Question 1) GCD=12 LCM= 252
The product of these two numbers is = 3024
Question 2) I think to find this you divide the product by the lcm because when i did it with question 1 it worked. So if this is how you do this, the answer would be 1/4. I dont see how thats possible but yeahhhh. But if you just divide the larger number by the smaller number, this answer would come out to be 4.
Question 1) They both find common numbers that are related to both a and b. The lcm can be divided by both a and b. The gcd can go into both a and b.
Question 2) I think the four most important numbers are 0, 1, 2, and 5. 0 because anytime you divide by 0 or multiply by 0, it will end up being no solution or 0. 1 is important because 1 can go into any number at all. When you divide a number by 1 it ends up being that number. Same when you divide by 1. 2 is important because half the numbers in the world can be divided by 2, since 2 goes into all even numbers. 5 is important because when you are dealing with prime factorization, this number shows up a lot.
Set 2 concrete:
Question 1) GCD=12 LCM= 252
The product of these two numbers is = 3024
Question 2) I think to find this you divide the product by the lcm because when i did it with question 1 it worked. So if this is how you do this, the answer would be 1/4. I dont see how thats possible but yeahhhh. But if you just divide the larger number by the smaller number, this answer would come out to be 4.
10/10/10
SET1-ABSTRACT
1) The relationship between gcd and lcm is that both a and b are either multiples or divisor of each other.
2)I think the four most important number are 2,3,5 and 7 because they are the four smallest prime numbers and they for used in prime factorization often.
SET2- CONCRETE
1)GCD(36,84)=12
LCM(36,84)=252
PRODUCT(252*12)=3024
2)I took the product and divided it by the lcm and got 1/4? so lol it might not be right
1) The relationship between gcd and lcm is that both a and b are either multiples or divisor of each other.
2)I think the four most important number are 2,3,5 and 7 because they are the four smallest prime numbers and they for used in prime factorization often.
SET2- CONCRETE
1)GCD(36,84)=12
LCM(36,84)=252
PRODUCT(252*12)=3024
2)I took the product and divided it by the lcm and got 1/4? so lol it might not be right
10/10/10
Set 1, abstract:
The relationship between them is that they are both multiples or factors or divisors of each other, a and b.
Question 2.) If you were asked to state what you think to be the four most important numbers to all of mathematics. I'm not asking for your lucky number, or 42, or 1337, or any other quirky/"clever" response. I want to know, mathematically speaking, what you think the four most important numbers are. P.S., the answer is not 1,2,3, and 4. (Also, there are no right or wrong answers, and if you REALLY think it's 1,2,3,4, just justify your answer VERY well.)
I think the four most important numbers are 0, 1, 2, and 10. 0 because when you multiply or do anything with 0, you get zero, or nothing happens to the number you are working with. 1 because when you multiply by this you get an identity sort of answer, exactly. 2 because I seem always to be working with the number 2. You use 2 for evens and primes and stuff, and 2 is just cool lol. 10 because big numbers are "derived" from 10, like 100, 1000, 10000, etc. The power of 10 rule is also important I think, that's why I said 10 is an important number.
Set 2, concrete:
Question 1.) What is the lcm and gcd of the numbers 36 and 84? What is the product of 36 and 84?
(36, 84)
36: 2x2x3x3
2^2 x 3^2
84: 2x2x3x7
2^2 x 3 x 7
lcm: 2^2 x 3^2 x 7 = 252
gcd: 2^2 x 3 = 12
36*84 = 3024
Question 2.) If the product of two numbers, a and b, is 1024 and their lcm 4096, what must their gcd be?
Yaaa I honestly don't know how to do this with the information given...when this was on the test, I just left it blank lol sorry.
10/10/10 hahhah
Question 1) They will contain the same number which means they are multiples or divisors of a and b
Question 2) I think the four most important numbers are 0 2 3 5 what have we been doing in number theory this whole time? dealing with prime factorization, so these are the smallest prime numbers. 0 is important because anything divided or multiplied by 0 is either impossible or 0. 2 is important because its also even so you can tell if a number is even and use it in prime factorization. 3, because so many numbers are multiples of 3. the same thing with 5, you can automatically tell if a number is divisible by 5 by looking at the end number, wether it be 5 ir zero.
set 2
question 1)
the GCD is 12, the lcm is 252
multiply them together and you get
product=3024
question 2)
B-Rob showed us how to find this, but it came out as 1/4, never got an answer like that so let's go with that.
Question 2) I think the four most important numbers are 0 2 3 5 what have we been doing in number theory this whole time? dealing with prime factorization, so these are the smallest prime numbers. 0 is important because anything divided or multiplied by 0 is either impossible or 0. 2 is important because its also even so you can tell if a number is even and use it in prime factorization. 3, because so many numbers are multiples of 3. the same thing with 5, you can automatically tell if a number is divisible by 5 by looking at the end number, wether it be 5 ir zero.
set 2
question 1)
the GCD is 12, the lcm is 252
multiply them together and you get
product=3024
question 2)
B-Rob showed us how to find this, but it came out as 1/4, never got an answer like that so let's go with that.
10/10/10 Post
Set 1.
1) The relationship between the gcd and lcm of a & b is that the gcd will always be a divisor of the lcm.
2) I think the most important numbers in mathematics are 2, 3, 5, and 7. I think this because they are the four prime numbers, and you can do many things with them. With 2 you can tell if something is even or odd by dividing that number by 2 and seeing if there is a remainder. If there is one, then the number is odd. I also think 3, 5, and 7 are important because they are the next smallest primes and are used very often when dealing with prime factorization.
Set 2.
1) LCM[36,84] = 12
GCD[36,84] = 252
36 * 84 = 3024 = 12 * 252 = LCM*GCD
2) a*b = 1024
lcm[a,b] = 4096
gcd[a,b] = ?
lcm*gcd = product
gcd = product/lcm
gcd = 1024/4096
gcd = 1/4
I don't think that there can be a non-integer gcd, so I therefore think this is probably wrong.
Ryan Breaud
1) The relationship between the gcd and lcm of a & b is that the gcd will always be a divisor of the lcm.
2) I think the most important numbers in mathematics are 2, 3, 5, and 7. I think this because they are the four prime numbers, and you can do many things with them. With 2 you can tell if something is even or odd by dividing that number by 2 and seeing if there is a remainder. If there is one, then the number is odd. I also think 3, 5, and 7 are important because they are the next smallest primes and are used very often when dealing with prime factorization.
Set 2.
1) LCM[36,84] = 12
GCD[36,84] = 252
36 * 84 = 3024 = 12 * 252 = LCM*GCD
2) a*b = 1024
lcm[a,b] = 4096
gcd[a,b] = ?
lcm*gcd = product
gcd = product/lcm
gcd = 1024/4096
gcd = 1/4
I don't think that there can be a non-integer gcd, so I therefore think this is probably wrong.
Ryan Breaud
10/10/10
Set 1:
1.) The gcd and lcm usually have common multiples or one of the numbers is the divisor of the other. For example: gcd and lcm (24, 54)
*find the prime factorization first and you get that
24's prime factorization is 2^3(3)
54's prime factorization is 3^3(2)
LCM = 216
GCD = 6
*In this case, 6 is a divisor of 216
2.) I think the four most important numbers in mathematics are 0, 1, 2, and 3.
0 is important because often when it is involved in some problems, the answer may be 0 or possibly undefined. i.e. whenever 0 is involved, several things cancel out because of it. 1 is also important because it is a simple number with simple rules to it..as in if it were involved in addition or subtraction, it doesn't dramatically change the solution to a problem; whereas when it's involved in multiplication or division, it doesn't change the answer at all because when multiplying or dividing a # by 1, you get that #. 2 is also important because it is the the number that can go evenly into most numbers (well I think so); especially since 2 is in the majority of categories of numbers, i.e.--2 is an integer, a whole #, natural #, even, power of 2, prime..etc. Lastly, 3 is another important number because it aquires some of the same characteristics as 2...3 is an integer, whole #, natural #, prime, odd. Also 3 is used when cubing a number or cuberooting it, as 2 is used when you want to find half of something or when you want to square a # or square root it.
Set 2:
1.) (36, 84)
*To find the gcd and lcm, you first find the prime factorization of both numbers.
36's prime fact. is 2^2(3^2)
84's prime fact. is 2^2(3)(7)
*Using the prime factorizations you find that the GCD = 12 and LCM = 252
*The product of 36 and 84 (which is also the product of the GCD & LCM) is 3024
2.) Okay for this question, I have no idea what to do because I tested it out with the gcd and lcm you gave in question 1 and it worked (because you take the product and divide it by the lcm)..but for this one I got 1/4...So I'm thinking maybe you multiply 1/4 by 1024 and get that the gcd is 256? (that's most likely wrong, but I tried..)
1.) The gcd and lcm usually have common multiples or one of the numbers is the divisor of the other. For example: gcd and lcm (24, 54)
*find the prime factorization first and you get that
24's prime factorization is 2^3(3)
54's prime factorization is 3^3(2)
LCM = 216
GCD = 6
*In this case, 6 is a divisor of 216
2.) I think the four most important numbers in mathematics are 0, 1, 2, and 3.
0 is important because often when it is involved in some problems, the answer may be 0 or possibly undefined. i.e. whenever 0 is involved, several things cancel out because of it. 1 is also important because it is a simple number with simple rules to it..as in if it were involved in addition or subtraction, it doesn't dramatically change the solution to a problem; whereas when it's involved in multiplication or division, it doesn't change the answer at all because when multiplying or dividing a # by 1, you get that #. 2 is also important because it is the the number that can go evenly into most numbers (well I think so); especially since 2 is in the majority of categories of numbers, i.e.--2 is an integer, a whole #, natural #, even, power of 2, prime..etc. Lastly, 3 is another important number because it aquires some of the same characteristics as 2...3 is an integer, whole #, natural #, prime, odd. Also 3 is used when cubing a number or cuberooting it, as 2 is used when you want to find half of something or when you want to square a # or square root it.
Set 2:
1.) (36, 84)
*To find the gcd and lcm, you first find the prime factorization of both numbers.
36's prime fact. is 2^2(3^2)
84's prime fact. is 2^2(3)(7)
*Using the prime factorizations you find that the GCD = 12 and LCM = 252
*The product of 36 and 84 (which is also the product of the GCD & LCM) is 3024
2.) Okay for this question, I have no idea what to do because I tested it out with the gcd and lcm you gave in question 1 and it worked (because you take the product and divide it by the lcm)..but for this one I got 1/4...So I'm thinking maybe you multiply 1/4 by 1024 and get that the gcd is 256? (that's most likely wrong, but I tried..)
october 10 post
set 1:
1) the relationship between the lcm and gcd of two positive integers a & b, is that both numbers are either multiples or divisors of a & b.
2)I think one of the 4 most important numbers is 0. because when you do anything with 0, you usually end up with 0. Another important number is 1, because when multiplying or dividing by one, you end up with the OTHER number. Another important number is 2 because it is the first prime number, the first even number, you can get ANY even number if you multiply anything by 2. it is used for formulas to find every odd/even number. it probably is the MOST important. And I think another important number is 5. just because it seems like that would be the other number
set 2:
1) lcm=252... gcd=12... product=3024
2)i honestly don't know how to find the gcd with only the information given to me..
1) the relationship between the lcm and gcd of two positive integers a & b, is that both numbers are either multiples or divisors of a & b.
2)I think one of the 4 most important numbers is 0. because when you do anything with 0, you usually end up with 0. Another important number is 1, because when multiplying or dividing by one, you end up with the OTHER number. Another important number is 2 because it is the first prime number, the first even number, you can get ANY even number if you multiply anything by 2. it is used for formulas to find every odd/even number. it probably is the MOST important. And I think another important number is 5. just because it seems like that would be the other number
set 2:
1) lcm=252... gcd=12... product=3024
2)i honestly don't know how to find the gcd with only the information given to me..
Friday, October 8, 2010
Blog Prompt, October 8, 2010
Hey everyone, sorry about last week. I managed to completely forget to post a prompt.
Anyways, here's this week's. It should be relatively laid back and mostly opinionated:
All questions are required:
Set 1, abstract:
Question 1.) What is the relationship between the gcd and lcm of two positive integers, a and b?
Question 2.) If you were asked to state what you think to be the four most important numbers to all of mathematics. I'm not asking for your lucky number, or 42, or 1337, or any other quirky/"clever" response. I want to know, mathematically speaking, what you think the four most important numbers are. P.S., the answer is not 1,2,3, and 4. (Also, there are no right or wrong answers, and if you REALLY think it's 1,2,3,4, just justify your answer VERY well.)
Justify briefly.
Set 2, concrete:
Question 1.) What is the lcm and gcd of the numbers 36 and 84? What is the product of 36 and 84?
Question 2.) If the product of two numbers, a and b, is 1024 and their lcm 4096, what must their gcd be?
Anyways, here's this week's. It should be relatively laid back and mostly opinionated:
All questions are required:
Set 1, abstract:
Question 1.) What is the relationship between the gcd and lcm of two positive integers, a and b?
Question 2.) If you were asked to state what you think to be the four most important numbers to all of mathematics. I'm not asking for your lucky number, or 42, or 1337, or any other quirky/"clever" response. I want to know, mathematically speaking, what you think the four most important numbers are. P.S., the answer is not 1,2,3, and 4. (Also, there are no right or wrong answers, and if you REALLY think it's 1,2,3,4, just justify your answer VERY well.)
Justify briefly.
Set 2, concrete:
Question 1.) What is the lcm and gcd of the numbers 36 and 84? What is the product of 36 and 84?
Question 2.) If the product of two numbers, a and b, is 1024 and their lcm 4096, what must their gcd be?
Sunday, October 3, 2010
Blog Prompt #4
Question 1:
1332
6x222
2x3 2x111
3x37
Answer: 2x3^2x37
Question 2:
240, 120x2, 60x2, 30x2, 15x2, 5x3 Prime Factorization= 2^4x3x5
840, 420x2, 210x2, 105x2, 21x5, 7x3 Prime Factorization= 2^3x3x5x7
GCD: 4
LCM: 1680
Question 3:
133 and 103 are both prime numbers. I added the factors together to see if anything can go into them, and then checked by regular division.
Question 4:
2= 1
3= 1
5= 0
13= 9
I did not feel like using the prime factorization for this method so i am not going to lie, I just did regular divison with some help from my calculator.
1332
6x222
2x3 2x111
3x37
Answer: 2x3^2x37
Question 2:
240, 120x2, 60x2, 30x2, 15x2, 5x3 Prime Factorization= 2^4x3x5
840, 420x2, 210x2, 105x2, 21x5, 7x3 Prime Factorization= 2^3x3x5x7
GCD: 4
LCM: 1680
Question 3:
133 and 103 are both prime numbers. I added the factors together to see if anything can go into them, and then checked by regular division.
Question 4:
2= 1
3= 1
5= 0
13= 9
I did not feel like using the prime factorization for this method so i am not going to lie, I just did regular divison with some help from my calculator.
Alaina's post, 3 Oct. 2010
Question 1:
What is the prime factorization of 1332?
1332->3, 444->2, 222->2, 111->3, 37
2*32*37
Question 2:
What is the gcd and lcm of 240 and 840?
GCD: 240->2, 120->2, 60->2, 30->2, 15->3, 5
prime factorization=24*3*5
840->2, 420-> 2, 210->2, 105->5, 21->3, 7
prime factorization=23*3*5*7
4*1*1=4
LCM:24*3*5*7
16*3*5*7=1680
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up)?
Both 133 and 103 are prime. You cannot find the prime factorization of either. None of the small primes go into either 133 or 103.
Question 4:
What is the remainder whe 4803925 is divided by 2? When it is divided by 3? By 5? By 13? How did you figure each out?
4803925/2= remainder of 1
4803925/3= remainder of 1
4803925/5= remainder of 0
4803925/13= remainder of 9
I really didn't feel like using the prime factorization method so I just divided. Finding the remainder when dividing by 2 and 5 were very easy. 5 goes into anything that ends in a 5 or 0 evenly. 2 goes into anything that is even so because it is odd, the remainder is 1. 3 goes into all of the numbers except 4803925 so i only divided 3 into 25-> remainder of 1. 13 was the hardest of them. it just required some simple long division.
What is the prime factorization of 1332?
1332->3, 444->2, 222->2, 111->3, 37
2*32*37
Question 2:
What is the gcd and lcm of 240 and 840?
GCD: 240->2, 120->2, 60->2, 30->2, 15->3, 5
prime factorization=24*3*5
840->2, 420-> 2, 210->2, 105->5, 21->3, 7
prime factorization=23*3*5*7
4*1*1=4
LCM:24*3*5*7
16*3*5*7=1680
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up)?
Both 133 and 103 are prime. You cannot find the prime factorization of either. None of the small primes go into either 133 or 103.
Question 4:
What is the remainder whe 4803925 is divided by 2? When it is divided by 3? By 5? By 13? How did you figure each out?
4803925/2= remainder of 1
4803925/3= remainder of 1
4803925/5= remainder of 0
4803925/13= remainder of 9
I really didn't feel like using the prime factorization method so I just divided. Finding the remainder when dividing by 2 and 5 were very easy. 5 goes into anything that ends in a 5 or 0 evenly. 2 goes into anything that is even so because it is odd, the remainder is 1. 3 goes into all of the numbers except 4803925 so i only divided 3 into 25-> remainder of 1. 13 was the hardest of them. it just required some simple long division.
Monday, September 27, 2010
Post # 4
soo..Prime factorization of 1332:
13322 * 6662 * 9 * 742 * 3 * 3 * 2 * 372^2 * 3^2 * 37
GCD and LCM of 240 and 840.
240 10 * 245 *
GCD= 2^3 * 3 * 5 = 120
LCM = 2^4 * 3 * 5 * 7 = 1680
If you divide both prime numbers starting with the smallest prime number and continuing to increase the number, you will realize that 19133 isn't prime; but 103 is.
The biggest number that thirteen is divisable by is 480316; since that is nine numbers less than a number divisable by it, the remainder is nine.
and that's about all i can do on this blog.
i'm not sure if i should know how to do this stuff..
or look it up. or maybe i'm just not thinkging properly.
13322 * 6662 * 9 * 742 * 3 * 3 * 2 * 372^2 * 3^2 * 37
GCD and LCM of 240 and 840.
240 10 * 245 *
GCD= 2^3 * 3 * 5 = 120
LCM = 2^4 * 3 * 5 * 7 = 1680
If you divide both prime numbers starting with the smallest prime number and continuing to increase the number, you will realize that 19133 isn't prime; but 103 is.
The biggest number that thirteen is divisable by is 480316; since that is nine numbers less than a number divisable by it, the remainder is nine.
and that's about all i can do on this blog.
i'm not sure if i should know how to do this stuff..
or look it up. or maybe i'm just not thinkging properly.
Sunday, September 26, 2010
Connor's blog 4
(Please show your work (briefly))
Question 1:
What is the prime factorization of 1332?
Question 2:
What is the gcd and lcm of 240 and 840?
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
Q1: 1332
2 x 666 ZOMG 666, THE NUMBER OF THE BEAST :O
2 x 333
3 x 111
3 x 37
2^2 x 3^2 x 37
Q2: GCD and LCM 240 and 840
GCD: 840-240=600
600- 240=360
360-240=120
240-120=120
LCM: 2^4 x 3 x 5 x 7 = 1680
Q3: 133 isn't prime; because I'm good like that and I checked it with numberrsss
103 is prime; same as the above reason
Q4: 4803925/2 = 2401962.5
4803925/3 = 1601308.3(repeating)
4803925/5 = 960785.0
4803925/13 = 369532.692
how did I figure them out? I used a calculator of course
Question 1:
What is the prime factorization of 1332?
Question 2:
What is the gcd and lcm of 240 and 840?
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
Q1: 1332
2 x 666 ZOMG 666, THE NUMBER OF THE BEAST :O
2 x 333
3 x 111
3 x 37
2^2 x 3^2 x 37
Q2: GCD and LCM 240 and 840
GCD: 840-240=600
600- 240=360
360-240=120
240-120=120
LCM: 2^4 x 3 x 5 x 7 = 1680
Q3: 133 isn't prime; because I'm good like that and I checked it with numberrsss
103 is prime; same as the above reason
Q4: 4803925/2 = 2401962.5
4803925/3 = 1601308.3(repeating)
4803925/5 = 960785.0
4803925/13 = 369532.692
how did I figure them out? I used a calculator of course
Connor's blog 4
(Please show your work (briefly))
Question 1:
What is the prime factorization of 1332?
Question 2:
What is the gcd and lcm of 240 and 840?
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
Q1: 1332
2 x 666 ZOMG 666, THE NUMBER OF THE BEAST :O
2 x 333
3 x 111
3 x 37
2^2 x 3^2 x 37
Q2: GCD and LCM 240 and 840
GCD: 840-240=600
600- 240=360
360-240=120
240-120=120
LCM: 2^4 x 3 x 5 x 7 = 1680
Q3: 133 isn't prime; because I'm good like that and I checked it with numberrsss
103 is prime; same as the above reason
Q4: 4803925/2 = 2401962.5
4803925/3 = 1601308.3(repeating)
4803925/5 = 960785.0
4803925/13 = 369532.692
how did I figure them out? I used a calculator of course
Question 1:
What is the prime factorization of 1332?
Question 2:
What is the gcd and lcm of 240 and 840?
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
Q1: 1332
2 x 666 ZOMG 666, THE NUMBER OF THE BEAST :O
2 x 333
3 x 111
3 x 37
2^2 x 3^2 x 37
Q2: GCD and LCM 240 and 840
GCD: 840-240=600
600- 240=360
360-240=120
240-120=120
LCM: 2^4 x 3 x 5 x 7 = 1680
Q3: 133 isn't prime; because I'm good like that and I checked it with numberrsss
103 is prime; same as the above reason
Q4: 4803925/2 = 2401962.5
4803925/3 = 1601308.3(repeating)
4803925/5 = 960785.0
4803925/13 = 369532.692
how did I figure them out? I used a calculator of course
Jeffers.!
1. Find the prime factorization of 1332
2 666
2 333
3 111
3 37
2^2 * 3^2 *37
2. GCD and LCM of 240 and 840
2 120
2 60
2 50
2 15
3 5
2^4 * 3*15
3. 840
2 420
2 210
2 105
5 21
3 7
2^3*5*7
GCD=2^3 * 3*5=120
LCM=2^4* 3*5*7= 1680
4.are 133 or 103 prime?
133 is not prime, it is divisible by 7 and 19.
103 is prime b/c it has no other factors beside 1 and itself.
You use a factor tree to solve this out, if you are unable to, then its prime
5. 4803925/2
I solved this by using loooong division.
2=1(odd number)
3=1(number adds to give you 31,one over 30 a multiple of 3)
5=0(goes into number evenly because 5 can go into any number evenly that ends in 0 or 5.)
13=9
2 666
2 333
3 111
3 37
2^2 * 3^2 *37
2. GCD and LCM of 240 and 840
2 120
2 60
2 50
2 15
3 5
2^4 * 3*15
3. 840
2 420
2 210
2 105
5 21
3 7
2^3*5*7
GCD=2^3 * 3*5=120
LCM=2^4* 3*5*7= 1680
4.are 133 or 103 prime?
133 is not prime, it is divisible by 7 and 19.
103 is prime b/c it has no other factors beside 1 and itself.
You use a factor tree to solve this out, if you are unable to, then its prime
5. 4803925/2
I solved this by using loooong division.
2=1(odd number)
3=1(number adds to give you 31,one over 30 a multiple of 3)
5=0(goes into number evenly because 5 can go into any number evenly that ends in 0 or 5.)
13=9
Justin 4
1. find the prime factorization of 1332
3 444
4 111
3 37
2^2x3^2x37
2. find the gcd and lcm of 240 and 840
240
3 80
8 10
2 5
2^4x3x5
840
7 120
5 24
3 8
2^3x3x5x7
GCD=2^3x3x5=120
LCM=2^4x3x5x7=1680
3. are 103 and 133 prime?
133 is not
103 is
divide by prime numbers to find out
4.remainder when 4083925 is diveded by 2, 3, 5, and 13
a. divide by 2 R=1
b. divide by 3 R=1 the digits add up to 31
c. divide by 5 R=0
d. divide by 13 R=9
3 444
4 111
3 37
2^2x3^2x37
2. find the gcd and lcm of 240 and 840
240
3 80
8 10
2 5
2^4x3x5
840
7 120
5 24
3 8
2^3x3x5x7
GCD=2^3x3x5=120
LCM=2^4x3x5x7=1680
3. are 103 and 133 prime?
133 is not
103 is
divide by prime numbers to find out
4.remainder when 4083925 is diveded by 2, 3, 5, and 13
a. divide by 2 R=1
b. divide by 3 R=1 the digits add up to 31
c. divide by 5 R=0
d. divide by 13 R=9
Heather's #4
1.) Prime Factorization of 1332:
1332
2(666)
2(333)
3(111)
3(37)
= 2^3 x 3^3 x 7
2.) GCD (240, 840)
*Prime factorizations:
240:
2(120)
2(60)
2(30)
2(15)
3(5)
=2^4 x 3 x 5
840:
2(420)
2(210)
2(105)
3(35)
5(7)
= 2^3 x 3 x 5 x 7
*To find the GCD, you look at the prime factorization of 240 and 840. You cross out the numbers they do not have in common (so for this one you would rule out 7). Then you multiply the numbers in common that have the least exponent.
*So you would take 2^3 x 3 x 5
GCD = 120
LCM (240, 840)
*For this you still use the prime factorization of both numbers but instead you multiply each number there with the highest exponent (and you include every number present, whether they have it in common or not)
*So you would get 2^4 x 3 x 5 x 7
LCM = 1680
3.) Is 133 prime? NO...Is 103 prime? YES!
*I simply used divisibility rules. And 133 is composite because I found that 7 can go into it evenly, whereas nothing can go into 103.
4.) Remainders when 4803925...
*is divided by 2 -- remainder is 1
*divided by 3 -- remainder is also 1
*divided by 5 -- there is NO remainder. 5 goes in evenly (divisibility rules say that 5 can go into any number that ends in a 5 or zero evenly)
*divided by 13 -- remainder is 9
**I found these out simply by dividing 4803925 by every number given, except 5.
1332
2(666)
2(333)
3(111)
3(37)
= 2^3 x 3^3 x 7
2.) GCD (240, 840)
*Prime factorizations:
240:
2(120)
2(60)
2(30)
2(15)
3(5)
=2^4 x 3 x 5
840:
2(420)
2(210)
2(105)
3(35)
5(7)
= 2^3 x 3 x 5 x 7
*To find the GCD, you look at the prime factorization of 240 and 840. You cross out the numbers they do not have in common (so for this one you would rule out 7). Then you multiply the numbers in common that have the least exponent.
*So you would take 2^3 x 3 x 5
GCD = 120
LCM (240, 840)
*For this you still use the prime factorization of both numbers but instead you multiply each number there with the highest exponent (and you include every number present, whether they have it in common or not)
*So you would get 2^4 x 3 x 5 x 7
LCM = 1680
3.) Is 133 prime? NO...Is 103 prime? YES!
*I simply used divisibility rules. And 133 is composite because I found that 7 can go into it evenly, whereas nothing can go into 103.
4.) Remainders when 4803925...
*is divided by 2 -- remainder is 1
*divided by 3 -- remainder is also 1
*divided by 5 -- there is NO remainder. 5 goes in evenly (divisibility rules say that 5 can go into any number that ends in a 5 or zero evenly)
*divided by 13 -- remainder is 9
**I found these out simply by dividing 4803925 by every number given, except 5.
Prompt 4 from Ryan
Prime factorization of 1332:
1332
2 * 666
2 * 9 * 74
2 * 3 * 3 * 2 * 37
2^2 * 3^2 * 37
GCD and LCM of 240 and 840.
240
10 * 24
5 * 2 * 2 * 12
5 * 2 * 2 * 6 * 2
5 * 2 * 2 * 3 * 2 * 2
2^4 * 3 * 5
840
10 * 84
5 * 2 * 2 * 42
5 * 2 * 2 * 2 * 21
5 * 2 * 2 * 2 * 3 * 7
2^3 * 3 * 5 * 7
GCD = 2^3 * 3 * 5 = 120
LCM = 2^4 * 3 * 5 * 7 = 1680
Are 133 or 103 prime?
You have to divide both by prime numbers starting with 2 and keep going until the number you are dividing by is bigger than the quotient.
133/7 = 19
133 is not prime
103 is prime.
R when 4803925 is divided by 2, 3, 5, and 13.
By 2, it will be R1 because it is an odd number.
By 3, it will be R1 because the number add to give you 31, which is one over a multiple of 3.
By 5, it will be R0 because 5 goes into the number evenly.
By 13, I found the biggest number 13 could go in to less than 4803925, and I got 480316. Since 480316 is 9 less than 4803925 then the remander is 9.
1332
2 * 666
2 * 9 * 74
2 * 3 * 3 * 2 * 37
2^2 * 3^2 * 37
GCD and LCM of 240 and 840.
240
10 * 24
5 * 2 * 2 * 12
5 * 2 * 2 * 6 * 2
5 * 2 * 2 * 3 * 2 * 2
2^4 * 3 * 5
840
10 * 84
5 * 2 * 2 * 42
5 * 2 * 2 * 2 * 21
5 * 2 * 2 * 2 * 3 * 7
2^3 * 3 * 5 * 7
GCD = 2^3 * 3 * 5 = 120
LCM = 2^4 * 3 * 5 * 7 = 1680
Are 133 or 103 prime?
You have to divide both by prime numbers starting with 2 and keep going until the number you are dividing by is bigger than the quotient.
133/7 = 19
133 is not prime
103 is prime.
R when 4803925 is divided by 2, 3, 5, and 13.
By 2, it will be R1 because it is an odd number.
By 3, it will be R1 because the number add to give you 31, which is one over a multiple of 3.
By 5, it will be R0 because 5 goes into the number evenly.
By 13, I found the biggest number 13 could go in to less than 4803925, and I got 480316. Since 480316 is 9 less than 4803925 then the remander is 9.
Feroz - Prompt 4
1. Prime factorization of 1332
1332
2 x 666
2 x 3 x 222
2 x 3 x 2 x 111
2 x 2 x 3 x 3 x 37
2. LCM of 240 and 840
240 = 2^4 x 3 x 5
840 = 2^3 x 3 x 7 x 5
= 120
3. Is 133 and 103 prime? And how do I know?
Well, 140 is divisible by 7. I subtracted 7 from 140 and got 133. So it's composite.
And for 103 I used the Sieve of Eratosthenes. And it's prime.
4. Remainder of 4803925 when divided by 2, 3, 5, and 13.
Any odd number divided by 2 has a remainder of 1. So 2: 1
For 3 I took the last two digits (25) and divided it by 3 ( 3 x 8 = 24). So 3: 1
It ends with a 5, so 5: 0
Unfortunately, I do not know a clever way for 13, so I used long division. 13: 9
1332
2 x 666
2 x 3 x 222
2 x 3 x 2 x 111
2 x 2 x 3 x 3 x 37
2. LCM of 240 and 840
240 = 2^4 x 3 x 5
840 = 2^3 x 3 x 7 x 5
= 120
3. Is 133 and 103 prime? And how do I know?
Well, 140 is divisible by 7. I subtracted 7 from 140 and got 133. So it's composite.
And for 103 I used the Sieve of Eratosthenes. And it's prime.
4. Remainder of 4803925 when divided by 2, 3, 5, and 13.
Any odd number divided by 2 has a remainder of 1. So 2: 1
For 3 I took the last two digits (25) and divided it by 3 ( 3 x 8 = 24). So 3: 1
It ends with a 5, so 5: 0
Unfortunately, I do not know a clever way for 13, so I used long division. 13: 9
Promp #4
Question 1:
What is the prime factorization of 1332?
(2,666)->(2,333)->(3,111)->(3,37)
22*32*37
Question 2:
What is the gcd and lcm of 240 and 840?
240=24*3*5; 840=23*3*5*7
gcd=23*3*5
gcd=120
lcm=24*3*5
lcm=840
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not?
Yes, 133 is prime, so is 103. When each is divided by one of the small primes, there is a remainder of 1, 3, or 13.
Question 4:
What is the remainder when 4803925 is divided by 2? What about 3? 5? and 13? How did you figure each out?
4803925/2 = 2401962 with a remainder of 1.
4803925/3 = 1601908 with a remainder of 1.
4803925/5 = 960785 with no remainder.
4803925/13 = 36958 with a remainder of 1.
What is the prime factorization of 1332?
(2,666)->(2,333)->(3,111)->(3,37)
22*32*37
Question 2:
What is the gcd and lcm of 240 and 840?
240=24*3*5; 840=23*3*5*7
gcd=23*3*5
gcd=120
lcm=24*3*5
lcm=840
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not?
Yes, 133 is prime, so is 103. When each is divided by one of the small primes, there is a remainder of 1, 3, or 13.
Question 4:
What is the remainder when 4803925 is divided by 2? What about 3? 5? and 13? How did you figure each out?
4803925/2 = 2401962 with a remainder of 1.
4803925/3 = 1601908 with a remainder of 1.
4803925/5 = 960785 with no remainder.
4803925/13 = 36958 with a remainder of 1.
Prompt 4
QUESTION 1:
1332
2 x 666
2 x 3 x 222
2 x 3 x 2 x111
2 x 2 x 3 x 3 x 37
QUESTION 2:
240
2 x 120
2 x 6 x 20
2 x 2 x 3 x 5 x 4
2 x 2 x 2 x 3 x 2 x 5
2^4 x 3 x 5
840
2 x 420
2 x 2 x 210
2 x 2 x 2 x 105
2 x 2 x 2 x 5 x 21
2 x 2 x 2 x 5 x 3 x 7
2^3 x 3 x 5 x 7
gcd= 2^3 x 3 x 5 = 120
lcm= 2^4 x 3 x 5 x 7 = 1680
QUESTION 3:
No, its divisible by 7 and 19
Yes, it is only divisible by 1 and itself.
To find then out you are going to figure out what each number is divisble by and if it is divisble by more numbers than 1 and itself then is composite, and if not then it is prime.
QUESTION 4:
I did long division for this.
2 = 1
3 = 1
5 = 0
13 = 9
1332
2 x 666
2 x 3 x 222
2 x 3 x 2 x111
2 x 2 x 3 x 3 x 37
QUESTION 2:
240
2 x 120
2 x 6 x 20
2 x 2 x 3 x 5 x 4
2 x 2 x 2 x 3 x 2 x 5
2^4 x 3 x 5
840
2 x 420
2 x 2 x 210
2 x 2 x 2 x 105
2 x 2 x 2 x 5 x 21
2 x 2 x 2 x 5 x 3 x 7
2^3 x 3 x 5 x 7
gcd= 2^3 x 3 x 5 = 120
lcm= 2^4 x 3 x 5 x 7 = 1680
QUESTION 3:
No, its divisible by 7 and 19
Yes, it is only divisible by 1 and itself.
To find then out you are going to figure out what each number is divisble by and if it is divisble by more numbers than 1 and itself then is composite, and if not then it is prime.
QUESTION 4:
I did long division for this.
2 = 1
3 = 1
5 = 0
13 = 9
Mary's Blog 4!
A)what is the prime factorization of 1332
1332
2x666
2x3x222
2x3x2x111
2x2x3x3x37
B)gcd(240,840)
240
2x120
2x6x20
2x2x3x5x4
2x2x2x3x2x5
2^4x3x5
840
2x420
2x2x210
2x2x2x105
2x2x2x5x21
2x2x2x5x3x7
2^3x3x5x7
gcd=2^3x3x5=120
lcm=2^4x3x5x7=1680
C) is 133 prime?
no, its divisible by 7 and 19
is 103 prime?
yes, it has no factors except 1 and itself.
so find this out, you attempt to make a factor tree, and if you can't, you know that its prime, or you can guess and check with a few numbers
D) 4803925/2
i just used regular old division on some paper for this
2=1
3=1
5=0 because it went evenly
13=9
1332
2x666
2x3x222
2x3x2x111
2x2x3x3x37
B)gcd(240,840)
240
2x120
2x6x20
2x2x3x5x4
2x2x2x3x2x5
2^4x3x5
840
2x420
2x2x210
2x2x2x105
2x2x2x5x21
2x2x2x5x3x7
2^3x3x5x7
gcd=2^3x3x5=120
lcm=2^4x3x5x7=1680
C) is 133 prime?
no, its divisible by 7 and 19
is 103 prime?
yes, it has no factors except 1 and itself.
so find this out, you attempt to make a factor tree, and if you can't, you know that its prime, or you can guess and check with a few numbers
D) 4803925/2
i just used regular old division on some paper for this
2=1
3=1
5=0 because it went evenly
13=9
Mal's Whatever post...
Question 1:
What is the prime factorization of 1332?
1332
2*666
2*2*333
2*2*3*111
2*2*3*3*37
2^2*3^2*37
Question 2:
What is the gcd and lcm of 240 and 840?
Prime factorization of each is:
240=2^4**3*5
840=2^3*3*7*5
so the gcd after taking all of the common numbers between the factorization and the lowest of those and multiplying it together(if that makes any sense) gives you:
gcd(240, 840)=120
now for lcm, you take the highest exponent between the two over every numbers...so this gives you:
lcm(240, 840)=1680
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Okay. so what I did for this thing is use the Sieve of Eran...something. So after writing the first couple of rows of the sieve starting from 101, I went through that process where you start with 2 and so on, crossing out all the multiples of prime numbers. Okay> so...the following I believe to be true:
133-composite...it gets canceled out by 7 in the sieve
103-prime. nothing goes into it.
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
So when I saw this, I was extremely tempted to just plug it in my calculator. That said, I did not.
I then began just dividing using long division (imagine that!). However, I do recognize that I can just find the prime factorization of it and put that over the number I'm dividing by. Again though, I relized that for say 5, I could just use my divisibility rules...5 obviously will go into a number evenly if it ends in what? 0 or 5 right? and this number does...so the remainders are as follows:
2:1
3:1
5:0
13:9
What is the prime factorization of 1332?
1332
2*666
2*2*333
2*2*3*111
2*2*3*3*37
2^2*3^2*37
Question 2:
What is the gcd and lcm of 240 and 840?
Prime factorization of each is:
240=2^4**3*5
840=2^3*3*7*5
so the gcd after taking all of the common numbers between the factorization and the lowest of those and multiplying it together(if that makes any sense) gives you:
gcd(240, 840)=120
now for lcm, you take the highest exponent between the two over every numbers...so this gives you:
lcm(240, 840)=1680
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Okay. so what I did for this thing is use the Sieve of Eran...something. So after writing the first couple of rows of the sieve starting from 101, I went through that process where you start with 2 and so on, crossing out all the multiples of prime numbers. Okay> so...the following I believe to be true:
133-composite...it gets canceled out by 7 in the sieve
103-prime. nothing goes into it.
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
So when I saw this, I was extremely tempted to just plug it in my calculator. That said, I did not.
I then began just dividing using long division (imagine that!). However, I do recognize that I can just find the prime factorization of it and put that over the number I'm dividing by. Again though, I relized that for say 5, I could just use my divisibility rules...5 obviously will go into a number evenly if it ends in what? 0 or 5 right? and this number does...so the remainders are as follows:
2:1
3:1
5:0
13:9
Helen's Blog #4
Question 1:
1332:
12x111
3x4x3x37
3x2x2x3x37
2²x3²x37
Question 2:
gcd(240,840)
240:
24x10
3x8x2x5
3x4x2x2x5
3x2x2x2x2x5
240= 2^4x3x5
840:
84x10
4x21x2x5
2x2x3x7x2x5
840= 2³x3x5x7
GCD = 2³x3x5= 120 LCM= 2^4x3x5x7=1680
Question 3:
133 is not a prime because you can divide it by 19 and 7. 103 is a prime because it is only divisible by itself.
Question 4:
Divide by 2: R=1
Divide by 3: R=1
Divide by 5: R=0
Divide by 9: R=9
I kept dividing until I couldn't divide anymore.
1332:
12x111
3x4x3x37
3x2x2x3x37
2²x3²x37
Question 2:
gcd(240,840)
240:
24x10
3x8x2x5
3x4x2x2x5
3x2x2x2x2x5
240= 2^4x3x5
840:
84x10
4x21x2x5
2x2x3x7x2x5
840= 2³x3x5x7
GCD = 2³x3x5= 120 LCM= 2^4x3x5x7=1680
Question 3:
133 is not a prime because you can divide it by 19 and 7. 103 is a prime because it is only divisible by itself.
Question 4:
Divide by 2: R=1
Divide by 3: R=1
Divide by 5: R=0
Divide by 9: R=9
I kept dividing until I couldn't divide anymore.
Kaitlyn's Blog #4
Question 1:
1332
3 x 444
3 x 148
2 x 74
2 x 37
The prime facorization of 1332 is: 2^2 x 3^2 x 37
Queston 2:
240
3 x 80
2 x 40
2 x 20
2 x 10
2 x 5
240= 2^4 x 3 x 5
840
3 x 280
2 x 140
2 x 70
2 x 35
5 x 7
840= 2^3 x 3 x 5 x 7
gcd (240,840)= 2^3 x 3 x 5= 120
lcm (240,840)= 2^4 x 3 x 5 x 7= 1680
Question 3:
133 is not prime, it is divisible by 7 and 19. 103 is prime because it can't be divided evenly by any number except 1 and itself.
Question 4:
2= 1
3= 1
5= 0
13= 9
I divided each of them on a piece and paper, and the last number that came out that i could not do anything with was the remainder.
1332
3 x 444
3 x 148
2 x 74
2 x 37
The prime facorization of 1332 is: 2^2 x 3^2 x 37
Queston 2:
240
3 x 80
2 x 40
2 x 20
2 x 10
2 x 5
240= 2^4 x 3 x 5
840
3 x 280
2 x 140
2 x 70
2 x 35
5 x 7
840= 2^3 x 3 x 5 x 7
gcd (240,840)= 2^3 x 3 x 5= 120
lcm (240,840)= 2^4 x 3 x 5 x 7= 1680
Question 3:
133 is not prime, it is divisible by 7 and 19. 103 is prime because it can't be divided evenly by any number except 1 and itself.
Question 4:
2= 1
3= 1
5= 0
13= 9
I divided each of them on a piece and paper, and the last number that came out that i could not do anything with was the remainder.
post 4
1- 1332 -2^2 x 3^2 x 37
2-666
2-333
3-111
3-37
2. (240,840) gcd = 120 ..... lcm = 1680
gcd-prime factorization multiply all common numbers with least exponent
lcm-prime factorization multiply all numbers with highest exponent
3.133 isn't prime, it is divisible by 7.(it may divisible by more i just stopped there) i just did guess & check.
103 i think is prime because you check all the numbers up to half way way.. which in this case would be 51.5
4.the remainder is one when divided by two.
the remainder is two when divided by three.
the remainder is zero when divided by five.
the remainder is nine when divided by thirteen.
you find the multiple of the number given before the number you are dividing the given number into... and find the difference
2-666
2-333
3-111
3-37
2. (240,840) gcd = 120 ..... lcm = 1680
gcd-prime factorization multiply all common numbers with least exponent
lcm-prime factorization multiply all numbers with highest exponent
3.133 isn't prime, it is divisible by 7.(it may divisible by more i just stopped there) i just did guess & check.
103 i think is prime because you check all the numbers up to half way way.. which in this case would be 51.5
4.the remainder is one when divided by two.
the remainder is two when divided by three.
the remainder is zero when divided by five.
the remainder is nine when divided by thirteen.
you find the multiple of the number given before the number you are dividing the given number into... and find the difference
Blog #4 Stephen
YAY, FALCONS WON THE GAME!! :) jk.
Anyway. Blog..blog...yes, blog.
Question 1:
What is the prime factorization of 1332?
What is the prime factorization of 1332?
1332: 148 x 9
2x74x9
2x2x37x3x3
2^2 x 3^2 x 37
Question 2:
What is the gcd and lcm of 240 and 840?
What is the gcd and lcm of 240 and 840?
gcd (240, 840)
240: 2x120
2x2x60
2x2x2x30
2x2x2x2x15
2x2x2x2x3x5
240: 2^4 x 3 x 5
840: 3x280
3x2x140
3x2x2x70
3x2x2x2x35
3x2x2x2x5x7
840: 2^3 x 3 x 5 x 7
gcd: 2^3 x 3 x 5 = 120
lcm: 2^4 x 3 x 5 x 7 = 1680
Question 3:
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
Is 133 prime? What about 103? How did you find out that it is/is not (without looking it up.)
133 is not prime because it is divisible by 7 and 19. 103 is prime because its only factors are 1 and 130.
Question 4:
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
What is the remainder when 4803925 is divided by 2? What is it when it is divided by 3? By 5? By 13?
How did you figure each of these out?
by 2 = 1
by 3 = 1
by 5 = 0
by 13 = 9
I used the remainder theorem. Divide regularly, then take the integer part and multiply by the divisor. Then subtract that from the original number, and you get the remainder.
Example: Remainder when 13 is divided by 7.
13/7 = 1.857
1 x 7 = 7
13 - 7 = 6
the remainder is 6
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