Bunuel
If n is a positive integer, is n(n^2 - 1) divisible by 42?
(1) n^3 is divisibly by 8.
(2) n is divisibly be 21.
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\(n(n^2-1)\) = n(n+1)(n-1)
n(n+1)(n-1) represents the product of three consecutive integers. Hence \( n(n+1)(n-1)\) will always be divisible by 6.
Question: Is n(n^2 - 1) divisible by 42 ?
42 = 7 * 3 * 2
As \( n(n+1)(n-1)\) will always be divisible by 6, we need to determine if \( n(n+1)(n-1)\) is divisible by 7.
Target Question: Is \( n(n+1)(n-1)\) divisible by 7?
Statement 1(1) \(n^3\) is divisibly by 8.While we can infer that n is even, we can't say for sure that n contains a 7. In other words, the statement is not sufficient to conclude whether 7 is a factor of n.
Case 1: If n is a multiple of 7, then the answer to the target question is Yes.
Case 2: If n is not a multiple of 7, then the answer to the target question is No.
As we are getting two answers to the target question, the statement alone is not sufficient. Eliminate A and D.
Statement 2(2) \(n\) is divisibly be 21.If n is divisible by 21, then n must be a multiple of 7. Put otherwise, 7 is a factor of n.
The statement is enough to answer a definite Yes to the target question - "
Is \( n(n+1)(n-1)\) divisible by 7?"
Option B