This isnt a GMAT question. On GMAT you dont have to select multiple answers. But its a good question.

Anyways, here you go..

See if x is divisible by 144 (2^4*3^2) that means 2^4*3^2 a is definitely factor of x. We are also told that cubed root of X is an integer. This means that X is a perfect cube. In order to be a perfect cube, powers (in this case 4 and 2) of the prime factors(in this case 2 and 3) of X must be divisible by 3. 4 and 2 are not divisible by 3. To make them divisible by 3, we increase 4 by 2, that gives are 6. Similarly we increase 2 by 1, that give us 3.

So one of the possible values of X could be 2^6*3^3 = 1728. Observe that 1728 is a cube of 12. So cube root of 1728 is 12 and 12 is divisible by 4 and 12. Thats the answer.

Hope that helps...

Yeah this question was on one of the MGMAT flashcards for reviewing concepts. Thanks so much for breaking it down and explaining this - makes a lot of sense now!

Other way (no prime factorizing and no multiplication):

x is divisble by 144. Thus, the smallest value of x is 12*12.

The cube root of x is an integer. Thus, x is a perfect cube. Thus, the smallest possible value now is 12*12*12.

The cube root of 12*12*12 is 12.

12 is divisble by 12 and by 4 but not by 8 and not by 9.