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1. LCM contains prime number 5, so m or n or both also contain 5. 2. if m:n=3:4 then only both m and m contain 5. Therefore, GCD is 5 or 10. 3. LCM contains 2^3, but in ratio m:n we have only 4=2^2. So, both m and m contain 2. GCD=10.

second way:

m*n=LCM*GCD - it is a formula. m*n=3x*4x=120*GCD ---> GCD=x^2/10 ---> only 10 works.
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Re: If the least common multiple of a positive integer [#permalink]

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13 Jun 2008, 06:45

walker wrote:

D

fist way:

LCM=120=3*2^3*5

1. LCM contains prime number 5, so m or n or both also contain 5. 2. if m:n=3:4 then only both m and m contain 5. Therefore, GCD is 5 or 10. 3. LCM contains 2^3, but in ratio m:n we have only 4=2^2. So, both m and m contain 2. GCD=10.

I like this. Thanks !

walker wrote:

second way:

m*n=LCM*GCD - it is a formula. m*n=3x*4x=120*GCD ---> GCD=x^2/10 ---> only 10 works.

Thanks for the refresh on the formula, I did not remember.

Re: If the least common multiple of a positive integer m and n [#permalink]

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20 Oct 2013, 16:04

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vd wrote:

If the least common multiple of a positive integer m and n is 120 and m:n is 3:4 what is the greatest common factor of m and n

3 5 6 10 12

Please provide detailed explanations on how to solve this

many thanks

I just plugged in numbers for m & n. If the ratio is 3:4, then using 3 & 4 for their values works fine, since we're trying to find the GCF, and not the actual values of m&n,

Re: If the least common multiple of positive integer m and n is [#permalink]

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12 Jan 2015, 01:46

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There are a number of different ways to approach this question. Given the "restrictions" that are in the prompt, if you're not sure what to do with a question such as this, you can always "brute force" it....

'Brute Force' is essentially just throwing numbers at a situation until you find the correct answer (or at least find the pattern that will lead you to the correct answer). It's not particularly elegant, but in the right circumstances it can be a really fast way to get to the correct answer.

In this question, we're told: 1) M and N are positive integers 2) The LCM of M and N is 120 3) The ratio of M:N is 3:4

I'm going to focus on how the second and third "restrictions" interact....

If M=3 and N=4, then the LCM would be 12 (not 120). Notice the "times 10" difference?....

What if... M = 30 and N = 40. Multiples of 30: 30, 60, 90, 120 Multiples of 40: 40, 80, 120 The LCM IS 120.

With 30 and 40 as our two values, it's not hard to find the GREATEST common factor. It has to be 10.

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