Let $p$ be a prime.

State and prove Lagrange's theorem on the number of solutions of a polynomial congruence modulo $p$. Deduce that $(p-1) ! \equiv-1 \bmod p$.

Let $k$ be a positive integer such that $k \mid(p-1)$. Show that the congruence

$$x^{k} \equiv 1 \quad \bmod p$$

has precisely $k$ solutions modulo $p$.