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    OlympiadHardNumber Theory7–9

    Divisibility Proof

    Prova de Divisibilidade

    Prove that for any positive integer nn, the expression n3−nn^{3} - n is divisible by 6.

    Factor n3−nn^{3} - n as n(n−1)(n+1)n(n-1)(n+1). These are three consecutive integers.

    Solution

    Step 1 of 6

    1. 1.n3−n=n(n2−1)=(n−1)⋅n⋅(n+1)n^{3} - n = n(n^{2} - 1) = (n-1) \cdot n \cdot (n+1)