Question 42
Let $p, q$ be prime numbers and $a$ be an integer such that $p \gt 2$ and $a \neq 1 \pmod{q}$ but $a^p \equiv 1 \pmod{q}$. Prove that
$$
(1+a^1)(1+a^2)\dots(1+a^{p-1}) \equiv 1 \pmod{q}.
$$
$$
(1+a^1)(1+a^2)\dots(1+a^{p-1}) \equiv 1 \pmod{q}.
$$