Divisibility of Odd Polynomial

CAT 1996 Slot 1 · QA · Medium · QA

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If nn is any odd number greater than 1, then n(n2−1)n(n^2 - 1) is always divisible by:

  1. A.

    divisible by 96 always

  2. B.

    divisible by 48 always

  3. C.

    divisible by 24 always

  4. D.

    None of these

Answer

C

Explanation

n(n2−1)=(n−1)n(n+1)n(n^2 - 1) = (n-1)n(n+1). Since nn is odd, (n−1)(n-1) and (n+1)(n+1) are consecutive even numbers, one divisible by 4, the other by 2   ⟹  \implies product divisible by 8. Product of 3 consecutive integers is divisible by 3. Thus divisible by 8×3=248 \times 3 = 24.

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