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.
Sunday, October 10, 2010
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.
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