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

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

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12 Jun 2008, 23:09
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If the least common multiple of positive integer m and n is 120, and m:n=3:4, what is the greatest common factor of m and n?

(A) 3
(B) 5
(C) 6
(D) 10
(E) 12
[Reveal] Spoiler: OA

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Last edited by Bunuel on 20 Oct 2013, 22:12, edited 2 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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Re: If the least common multiple of a positive integer [#permalink]

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12 Jun 2008, 23:26
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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]

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13 Jun 2008, 03:56
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]

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13 Jun 2008, 05:37
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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]

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13 Jun 2008, 05:45
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]

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13 Jun 2008, 05:48
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]

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13 Jun 2008, 05:53
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 m and n [#permalink]

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20 Oct 2013, 15:04
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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,

so

m*n=LCM*GCF

3*4=120*GCF
12=120*GCF
10=GCF

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

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

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12 Jan 2015, 15:29
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Hi All,

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.

Final Answer:
[Reveal] Spoiler:
D

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

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12 Jan 2015, 22:41
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m:n = 3:4

$$m = \frac{120}{4} = 30$$

$$n = \frac{120}{3} = 40$$

GCD of 30 & 40 = 10

Answer = D
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If the least common multiple of positive integer m and n is [#permalink]

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20 Sep 2015, 23:25
120=5*2*3*2*2

3/4 means 3 and 2*2 in two numbers exist, so we should give another 2*5 to both numbers to get the same ratio

3*5*2/2*2*2*5=30/40

GCF=10

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

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14 Mar 2016, 04:28
vd wrote:
If the least common multiple of positive integer m and n is 120, and m:n=3:4, what is the greatest common factor of m and n?

(A) 3
(B) 5
(C) 6
(D) 10
(E) 12

Supposing that m and n are 3*x and 4*x; LCM of m and n = 12*x= 120=>x=10
Since 3 and 4 are co-prime numbers 10 must be the gcf.
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Re: If the least common multiple of positive integer m and n is [#permalink]

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26 Nov 2016, 06:57
Nice Question.
Here is what i did =>
As m/n=3/4
Let m=3x
n=4x

Now GCD = Common elements => x
LCM=common elements *leftovers => x*3*4=> 12x

Hence !2x=120
so x=10
Hence GCD=10
Hence D
Additionally we can say that numbers m and n must be 30 and 40
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Re: If the least common multiple of positive integer m and n is   [#permalink] 26 Nov 2016, 06:57
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