C wrote:If the LCM of a and 12 is 36, what are the possible values of a?

Please explain.

Hi,

in future, please always provide 2 things:

1) all the answer choices; and
2) the source of the problem.

The answer choices are important because they provide valuable insight into how to approach the problem. For example, we could attack this question very quickly by looking at the choices and using process of elimination.

The source is important because we need to know if this is a good approximation of a real GMAT question. If the source is reliable, then we should take the question seriously; if the source isn't reliable, then we may be wasting our time on concepts that are irrelevant to the exam.

To address the specific question:

"LCM" stands for "lowest common multiple" (of course on the actual test, you'll never see the abbreviation, which already makes me question the source - or perhaps you just didn't provide the full question, which isn't optimal). The lowest common multiple of two integers is the smallest positive integer into which they both divide.

There are a number of ways we can solve this question, but the quickest is likely to list all the factors of 36:

1, 36
2, 18
3, 12
4, 9
6

We can quickly eliminate any numbers that are factors of 12, since the LCM of 12 and a factor of 12 would be 12.

So, cross off 1, 2, 3, 4, 6 and 12.

That leaves 9, 18 and 36.

Other than 12 and 36, the only other multiple of 12 small enough is 24. Since none of 9, 18 or 36 are factors of 24, each of them shares with 12 a LCM of 36.

So, the possible values of a are 9, 18 and 36.

Note that it's also possible to solve by using prime factors:

the prime factors of 12 are 2, 2 and 3.

The prime factors of 36 are 2, 2, 3 and 3.

So, any factor of 36 that contains 2 "3"s won't have any smaller common multiples with 12.

Accordingly, the numbers that share with 12 36 as a LCM are:

3*3 = 9;
2*3*3 = 18; and
2*2*3*3 = 36.