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Math Help - Divisibility (gcd) 9

  1. #1
    Sea
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    Divisibility (gcd) 9

    n \geq 2 and  k \in \mathbf{Z^{+}}


    Show that:

    (n-1)^{2}|n^{k}-1 \Leftrightarrow n-1|k
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  2. #2
    Senior Member JaneBennet's Avatar
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    This is the same as saying that n^2\mid (n+1)^k-1\ \Leftrightarrow\ n\mid k for all n,k\in\mathbb{Z}^+.

    Note that (n+1)^k-1\equiv kn\pmod{n^2}.

    Hence n^2\mid(n+1)^k-1\ \Leftrightarrow\ n^2\mid kn\ \Leftrightarrow\ n\mid k.
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