Introduction to P.M.I

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Question

For all odd positive integer $n,$ the number $n (n^{2} - 1)$ is divisible by

Moderate

Solution

For $n = 1,$ we have
$n (n^{2} - 1) = 0$ which is divisible by 24.
For $n = 3,$ we have
$n (n^{2} - 1) = 3 \times (9 - 1) = 24$ which is divisible by 24.
For $n = 5,$ we have
$n (n^{2} - 1) = 3 \times (25 - 1) = 72,$ which is divisible by 24.
Hence, $n (n^{2} - 1)$ is divisible by 24.

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