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Math Help - Prime Divisability

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    MHF Contributor chiph588@'s Avatar
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    Prime Divisability

    Let p be an odd prime number. Prove that the numerator of 1+1/2+1/3+...+1/(p-1) (when expressed as a single fraction with denominator (p-1)!) is divisible by p.
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    Quote Originally Posted by chiph588@ View Post
    Let p be an odd prime number. Prove that the numerator of 1+1/2+1/3+...+1/(p-1) (when expressed as a single fraction with denominator (p-1)!) is divisible by p.
    Let N be numerator.

    Then, N\equiv \sum_{k=1}^{p-1} p! (k)^{-1} (\bmod p). Where (k)^{-1} is inverse mod p.

    But \sum_{k=1}^{p-1} p! (k)^{-1} = p! \sum_{k=1}^{p-1}k.
    Since (k)^{-1} is a permuation of \{1,2,...,p-1\}.

    Finally, \sum_{k=1}^{p-1} k = \frac{p(p-1)}{2}.
    Therefore, it is divisible by p.
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