GRE test prep question [LCM and divisors] 
Let $S$ be the set of all positive integers $n$ such that $n^2$ is a
multiple of both $24$ and $108$.  Which of the following integers are
divisors of every integer $n$ in $S$ ?
Indicate all such integers:
$A:12$
$B:24$
$C:36$
$D:72$

The answers are $A$ and $C$
First I took the lcm of $24$ and $108$ which is $2^3\times3^3$ but then it says that "the prime factorization of a square number must contain only even exponents.  Thus, the least multiple of $(2^3)(3^3)$ that is a square is $(2^4)(3^4)$"
Can somebody explain why that is true?
What if the lcm was $2^3\times3^4$ ? Would I just make it $2^4\times3^4$ ?
Help!

We know that solution set is multiple of LCM . But here n^2 must be multiple of LCM . For that to happen it can be the number must be a perfect square . That can happen only if the power is even . Or else we will end up with square root in n .
In your case $ 2^3 * 3^4 $ as $n^2$. n will have $\sqrt{2}$ which makes n a non integer . so the numbers must be 216*6 , 216*6^3 ,...
