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divisible by 12?

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by sanju09 » Mon Mar 26, 2012 1:04 am
If n is a positive integer greater than 1, is 3n(n^2 − 1) divisible by 12?
(1) n is odd.
(2) n is a multiple of 3.
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by Anurag@Gurome » Mon Mar 26, 2012 1:57 am
sanju09 wrote:If n is a positive integer greater than 1, is 3n(n^2 − 1) divisible by 12?
(1) n is odd.
(2) n is a multiple of 3.
(1) n is odd.
Let us assume that n = 2k + 1 for any positive integer k
3n(n² − 1) = 3(2k + 1)[(2k + 1)² - 1] = 3(2k + 1)[4k² + 4k + 1 - 1] = 3 * 4k(2k + 1)[k + 1] = 12k(2k + 1)[k + 1], which is clearly divisible by 12; SUFFICIENT.

(2) n is a multiple of 3.
Let us assume that n = 3k for any positive integer k
3n(n² − 1) = 3(3k)[(3k)² - 1] = 9k[9k² - 1], which may or may not be divisible by 12; NOT sufficient.

The correct answer is A.
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