sana.noor wrote:If p is a positive integer and p^2 is divisible by 12, then the largest positive integer that must divide p^3 is
(A) 2^3
(B) 2^6
(C) 3^3
(D) 6^3
(E) 12^2
OA is D
The correct answer choice MUST divide p³ -- even if p³ is its MINIMUM POSSIBLE VALUE.
Since p is an integer, p² is a perfect square:
1, 4, 9, 16, 25,
36...
Of these values, the least possible multiple of 12 is 36.
If p²=36, then p=6 and p³=6³.
Thus, the minimum possible value for p³ is 6³.
Clearly, 6³ is divisible by answer choice
D -- the greatest of the five answer choices.
The correct answer is
D.
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