If p is prime, prove that for any integer a, p divides a^p + (p-1)!a and p divides (p-1)!a^p + a
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For a prime p Wilson's theorem says that (p-1)! = -1 mod p. Also, Fermat's little theorem says that for a s.t. (a,p) = 1 (no common factor), a^(p-1) = 1 (p), which implies that a^p = a (p). Try these in your problem.
Note that $\displaystyle a^p \equiv a \mod{p} \; \forall a $ though.
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