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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] New post   Updated on: 20 Oct 2013, 23:12
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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
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D

Originally posted by vd on 13 Jun 2008, 00:09.
Last edited by Bunuel on 20 Oct 2013, 23:12, 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] New post   13 Jun 2008, 06:37
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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 positive integer m and n is [#permalink] New post   12 Jan 2015, 16:29
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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:
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D


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Re: If the least common multiple of a positive integer [#permalink] New post   13 Jun 2008, 00:26
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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] New post   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] New post   13 Jun 2008, 06:45
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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] New post   13 Jun 2008, 06:48
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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] New post   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 m and n [#permalink] New post   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,

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] New post   12 Jan 2015, 23:41
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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] New post   21 Sep 2015, 00:25
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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] New post   14 Mar 2016, 05:28
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] New post   26 Nov 2016, 07:57
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] New post   07 Jul 2018, 18:33
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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


We can let m = 3x and n = 4x for some positive integer x. We see that since 3 and 4 have no common factor (other than 1), the least common multiple (LCM) of m and n must be (3)(4)(x). Since it’s given that the LCM is 120, we have:

(3)(4)(x) = 120

12x = 120

x = 10

So m = 30 and n = 40, and we see that 10 is the greatest common factor of m and n.

Answer: D
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Re: If the least common multiple of positive integer m and n is [#permalink] New post   19 Dec 2019, 03:03
ScottTargetTestPrep wrote:
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


We can let m = 3x and n = 4x for some positive integer x. We see that since 3 and 4 have no common factor (other than 1), the least common multiple (LCM) of m and n must be (3)(4)(x). Since it’s given that the LCM is 120, we have:

(3)(4)(x) = 120

12x = 120

x = 10

So m = 30 and n = 40, and we see that 10 is the greatest common factor of m and n.

Answer: D


wd it be correct to say that 10 is the LOWEST gcf of m,n?

in other words, is it correct to assume that given the restrictions on m,n, there cd be other values of m,n that can give an lcm of 120 and have 10 as the gcf?
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Re: If the least common multiple of positive integer m and n is [#permalink] New post   20 Dec 2019, 07:39
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Mansoor50 wrote:

wd it be correct to say that 10 is the LOWEST gcf of m,n?

in other words, is it correct to assume that given the restrictions on m,n, there cd be other values of m,n that can give an lcm of 120 and have 10 as the gcf?


The short answer is no. There's only one pair of m,n which are in the ratio of 3 to 4 and which have a LCM of 120. Thus, 10 is The greatest common factor of m and n.

Any other pair m,n will either not have a ratio of 3 to 4 or will not have a LCM of 120. Since the ratio of m to n is 3:4, different values of m = 3x and n = 4x will be given by different values of x. Since the LCM of m and n is 120, both 120/3x = 40/x and 120/4x = 30/x are integers. Thus, x can only equal 1, 2, 5 or

10. Hence, the only possibilities for the pair m, n are 3 and 4, 6 and 8, 15 and 20, 30 and 40. Of these pairs, only 30 and 40 have a LCM of 120.
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Re: If the least common multiple of positive integer m and n is [#permalink] New post   21 Mar 2020, 20:56
why is LCM product of x and 'leftovers'? is this always true?
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Re: If the least common multiple of positive integer m and n is [#permalink] New post   30 Mar 2020, 10:44
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


Asked: 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?

LCM (m,n) = 120 ; m = 3k , n= 4k
m*n = 3k * 4k = 12k^2 = 120 * GCF(m,n)
k = GCF (m,n)
12k^2 = 120 * k
k = 10 = GCF (m,n)

IMO D
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If the least common multiple of positive integer m and n is [#permalink] New post   22 Mar 2021, 13:50
Hi,

Got it wrong at first then here is how I tacked it:

LCM 120 = 12*10= 4*3*5*2 = 2^3*3*5

M:N = 3x:4x
My 3 must come from M and I have two 2's that must come from N therefore my third 2 must be in x, and because I have it in x it must be in M too.
The remaining integer, 5, is also in x so we have x= 5*2=10

M=30 and N=40, Therefore the GCF is 10.

Answer D)
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Re: If the least common multiple of positive integer m and n is [#permalink] New post   22 Jun 2022, 04:36
For any two numbers a and b,
LCM*HCF = a*b


Let the positive integer m be 3x.
Thus, n = 4x

Thus, the HCF of the numbers will be x.

So,
120*x = 3x*4x
120 = 12x
x = 12.

Thus, the correct option is E.
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Re: If the least common multiple of positive integer m and n is [#permalink]
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