Competition problem — Number theory

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Number theory Difficulty 5.8 AIME, harder Prove it

5. Let $p$ be a prime number, and $a$ be any integer. Prove:
(i) $p \mid a^{p}+(p-1)!a$ ;
(ii) $p \mid (p-1)!a^{p}+a$ .

Solution

5. Use Theorem 1 and Fermat's Little Theorem.

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