Problem Solving

LUANDATO wrote:What is the smallest positive integer x, such that 1,152x is perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

------ASIDE--------------------------------------
Let's examine a perfect CUBE.

1000 is a perfect cube since 1000 = 10^3
Now check out the prime factorization of 1000
1000 = (2)(2)(2)(5)(5)(5)
= [(2)(5)][(2)(5)][(2)(5)]
Notice that we can take the prime factorization of 1000 and divide the prime factors into THREE identical groups
-----NOW ONTO THE QUESTION------------------------

1,152 = (2)(2)(2)(2)(2)(2)(2)(3)(3)
So, 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(x)
Let's test the answer choices....

A. x = 4
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(4)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(2)
In order for the above to be a perfect CUBE, we must be able to divide the prime factors into THREE identical groups.
Since we cannot do that here, we can ELIMINATE A

B. x = 6
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(6)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(3)
Can we divide the above prime factors into THREE identical groups?
NO! ELIMINATE B

C. x = 8
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(8)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(2)(2)
Can we divide the above prime factors into THREE identical groups?
NO! ELIMINATE C

D. x = 12
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(12)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(2)(3)
Can we divide the above prime factors into THREE identical groups?
YES!
1152x = [(2)(2)(2)(3)][(2)(2)(2)(3)][(2)(2)(2)(3)]
So, if x = 12, then 1152x IS a perfect cube.

Answer: D