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# If the least common multiple of a positive integer m and n

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If the least common multiple of a positive integer m and n [#permalink]  13 Jun 2008, 00:09
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
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 00:26
vdhawan1 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

D for me!
x is the least common factor of m and n
m*n=3x*4x=120*x, so x=10, because x can not be 0!
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 04:56
sondenso wrote:
D for me!
x is the least common factor of m and n
m*n=3x*4x
I don't get this (where does it come from ?)

sondenso wrote:
3x*4x=120*x

This is just false

sondenso wrote:
=120*x, so x=10, because x can not be 0!

I don't get this either.

Can you explain ?
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 05:08
I m not sure that i understand the solution correctly
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 05:49
And I'm sure I don't
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 06:37
1
KUDOS
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.

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]  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.

But why is m*n = 3x*4x ?
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 06:48
Oski wrote:
But why is m*n = 3x*4x ?

m:n=3:4 --> m=3x, n=4x where x is an integer (m/n=3x/4x=3/4)
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 06:53
walker wrote:
Oski wrote:
But why is m*n = 3x*4x ?

m:n=3:4 --> m=3x, n=4x where x is an integer (m/n=3x/4x=3/4)

Yes, sure, but why is this x necessarily the GCD ?

Edit : Okay, I got it. This is because there is no common divisors in 3 and 4... (I guess this should be part of the demonstration ^^)
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Re: If the least common multiple of a positive integer [#permalink]  13 Jun 2008, 19:16
Oski wrote:
sondenso wrote:
D for me!
x is the least common factor of m and n
m*n=3x*4x
I don't get this (where does it come from ?)

sondenso wrote:
3x*4x=120*x

This is just false

sondenso wrote:
=120*x, so x=10, because x can not be 0!

I don't get this either.

Can you explain ?

Many thanks Walker, You have thorough explaination!
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Re: If the least common multiple of a positive integer   [#permalink] 13 Jun 2008, 19:16
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