Can someone answer this with explanation?

[lanefoly]

This question is from ManhattanGMAT, they provided an answer and explanation which I don't understand.

X is divisible by 144. If X^1/3 is an integer, then which of the following is X^1/3 definitely divisible by? (choose all that apply)
a) 4
b) 8
c) 9
d) 12
Note : X^1/3 is the cubic root of X.

Thanks in advance.

[nitingupta5]

the answer is 12

[el_torero]

here's my reasoning:

start by doing a prime factorization of 144: 144 = (2^4)*(3^2)

(1) if X is divisible by 144, then X = M*(2^4)*(3^2) for some positive integer M.
therefore, we're guaranteed that 2 and 3 are two of prime factors of X.

(2) if X^(1/3) is an integer, then X's prime factors must have exponents in multiples of 3.

combining the two pieces of information, we know that 2^6 and 3^3 must be prime factors of X.

so that means 2^2 and 3^1 are factors of X^(1/3), which implies 4 and 12 are factors of X. in fact, 2, 4, 6, 12 are all factors.

hope this helps.

[lanefoly]

Thanks el_torero. This is what I needed.