For positive integers M and N, if 24 is a divisor of M and 36 is a

Question

﻿For positive integers M and N, if 24 is a divisor of M and 36 is a divisor of N, what is the greatest integer that must be a divisor of M + N?

A. 12
B. 24
C. 36
D. 60
E. 72

Paras96 · Accepted Answer

﻿For positive integers M and N, if 24 is a divisor of M and 36 is a divisor of N, what is the greatest integer that must be a divisor of M + N?

Let a and b be positive integers

M = 24a
N = 36b

M+N=24a+36b=12(2a+3b)=12K

Therefore, in must case 12 is the greatest integer that must be a divisor of M + N

Hence A

Bunuel · Answer

So, you are basically asking why M and N cannot be 24 and 36, respectively. Notice that the question asks " what is the greatest integer that MUST be a divisor of M + N? " NOT " COULD be a divisor of M + N? ". While any number COULD be a divisor of M + N, only 12 is ALWAYS a divisor of M + N, regardless of what they are (provided 24 is a divisor of M and 36 is a divisor of N). For example, if M and N are 48 and 36, respectively, then among the options only 12 is a factor of M + N = 84.

AryaSwagat · Answer

Answer = Highest Common Factor of 24 and 36 = 12

Let M= 24 a = (2^3 * 3 ) a
and N= 36 b= (2^2*3^2) b
So M+N= 24 a+36 b = (2^3* 3)a+ (2^2*3^2) b = 2^2*3 (2a+3b)= 12 (2a+3b)

Answer 12 
 A

gmatophobia · Answer

M = 24x

N = 36y

M + N = 24x + 36y

= 12*(2x + 3y)

2x + 3y can either be odd or it can be even. Hence, we cannot take out any additional factors without knowing the value of  (2x + 3y) .

The sum however must be divisible by 12.

Option A

JeffTargetTestPrep · Answer

Since 24 is a divisor of M, we have:

M = 24m, where m is an integer.

Since 36 is a divisor of N, we have:

N = 36n, where n is an integer.

M + N = 24m + 36n = 12(2m + 3n)

We see that 12 is definitely a divisor of M + N.

What about 2m + 3n ? Besides 1, the expression 2m + 3n is not necessarily divisible by any other positive integer.

So, the greatest integer that MUST be a divisor of M + N is 12.

Answer: A

gautham83 · Answer

Why can't 'm' and 'n' be 1? 28 can be a divisor of 28 and 36 can be a divisor of 36. So the answer would be '60'?

Anonymous · Answer

So, since we're given the divisors of M,N and we're asked about the max possible divisor of M+N (unconditionally) then we're just looking for the greatest common factor of 24,36?
And similarly if we wanted to find the minimum integer that must be a divisor of the sum we would looking for the least common factor?
Finally, - if my reasoning above is correct - would it be true for every arithmetic operation or just for the sum?

Bunuel · Answer

Not followiong your logic.

The question does NOT ask about the greatest integer that COULD divide M + N. ANY number could be a divisor of M + N.

The question asks about the greatest integer that MUST be a divisor of M + N.

P.S. Also, note that the least common positive factor of two integers is 1.

Anonymous · Answer

I know that we're looking for the maximum integer that must be a divisor, that's why I added in parenthesis the unconditionally . What I'm saying is that we're given divisors of each one seperately and we want to find the greatest one of the sum that MUST be a divisor. Which lead me to the question if that means that we're looking for the greatest integer of both divisors that, no matter what is a divisor of the sum.

Then, I expanded my query to whether is the same logic for the minimum integer (if it had been asked) and also if the methodology can be used in any other arithmetic operation or is just for the sum.

Lastly, of course I would excluded the 1 if my reasoning is correct since as you said is the same for every integers.

GMATkid1997 · Answer

Is there any difference between these 2 questions?

GCF of (N + M)
GCF of (N , M)

They look the same, and appear to yield the same result, but I want to make sure I am 100% correct

Thanks in advance!

Bunuel

DanTheGMATMan · Answer

Find the greatest common factor of M and N, and M + N must be a multiple of that:

https://www.youtube.com/watch?v=bKBymKD8K0c

Bunuel · Answer

GCF stands for Greatest Common Factor, and it applies to two or more integers. For example, the GCF of 12, 16, and 20 is 4. The term GCF cannot be applied to a single number, so asking for the GCF of (N + M) doesn’t make sense. The question is actually about the greatest factor that the sum (N + M) must have.

Gambrosio · Answer

Prime factors:

M = 24k = 2^3x3^1*k
N = 36k = 2^2x3^2*k

greatest divisor: take the lowest powers of the common prime factors

2^2x3^1 = 12

Hence A

maingi95 · Answer

Why are the words 'divisor' and 'factor' used interchangeably? I noticed in other questions as well. When dividing a dividend, the divisor can yield a remainder, whereas in the case of a factor, the remainder must be zero. So in this case M can be 30 as well with divisor 24 and remainder 6, so by that logic all the options can be divisors of M + N. Let me know where is my understanding wrong?

Bunuel · Answer

In this case, divisor = factor.

00291681 · Answer

This was my first approach to the problem. Does it always works? I saw that people used a more simpler approach.

Krunaal · Answer

As mentioned here: https://gmatclub.com/forum/for-positive ... l#p3456397, we are asked to find a greatest number that must be a factor M + N; the approach you mentioned finds GCF of M and N, it's working here since when you add you bring out the greatest common factors, but it's worth noting the difference and having a clarity what the question asks.

bot00078 · Answer

So,
Here in this question i done is M/24 and N/36 but the thing mention is  greatest integer This one is indicate G.C.F.(Greatest Common Factor)
G.C.F. of (24,36) is 12 so the greatest integer is also 12

For positive integers M and N, if 24 is a divisor of M and 36 is a

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

For positive integers M and N, if 24 is a divisor of M and 36 is a

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards