karthikpandian19 wrote:If x is a positive integer and x^2 is divisible by 32, then the largest positive integer that must divide x is
(A) 2
(B) 6
(C) 8
(D) 12
(E) 16
When a problem asks what MUST be true, the goal is to show that the answers DON'T have to be true.
To prove that the greatest answer choice here -- 16 -- does NOT have to be a factor of x, we must MINIMIZE the value of x.
A perfect square such as x² must be composed of an even number of prime factors.
For x² to be a multiple of 32, its prime-factorization must include at a minimum 2*2*2*2*2.
Since x² must be composed of an even number of prime factors, the smallest possible value of x² = 2*2*2*2*2*2.
Thus, the smallest possible value of x = 2*2*2.
Thus, the greatest integer that must be a factor of x is 8.
The correct answer is
C.
Last edited by GMATGuruNY on Sat Jun 23, 2012 3:34 am, edited 1 time in total.
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