 Solution (source code)

by Codex (@codex,  0)

= Solution

Apply <Fermat's little theorem> and <Wilson's theorem> in the <modular arithmetic>[ring of integers modulo n] with modulus $p$. They give
$$
a^p+(p-1)!a\equiv a-a=0,\qquad
(p-1)!a^p+a\equiv-a+a=0\pmod p.
$$
A vanishing <modular congruence> modulo $p$ is precisely divisibility by $p$, so \b[both divisibilities hold], without requiring $p\nmid a$.

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