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Math Help - Working with mod p (Euler's and Fermat's Theorem)

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    Working with mod p (Euler's and Fermat's Theorem)

    Prove that if p is prime, then for any number a, divisible by p or not, a^p \equiv a (mod p)
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Zennie View Post
    Prove that if p is prime, then for any number a, divisible by p or not, a^p \equiv a (mod p)
    Are you asking to prove Euler's theorem? Do you know group theory? Note that \left(\mathbb{Z}/n\mathbb{Z}\right)^{\times} forms a group whose order is \varphi(n) and since any element of a group to it's order is the identity element the conclusion follows.

    If you're just trying to apply it merely notice that if a\nmid p then by virtue of p's primality (a,p)=1 in which case Fermat's theorem applies. If a\mid p then either a\equiv 0,1 in which case the theorem is a triviality.
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