20. a) Show that if $p$ is prime, then the only solutions of the congruence $x^{2} \equiv x(\bmod p)$ are those integers $x$ with $x \equiv 0$ or $1(\bmod p)$.
b) Show that if $p$ is prime and $k$ is a positive integer, then the only solutions of $x^{2} \equiv x\left(\bmod p^{k}\right)$ are those integers $x$ such that $x \equiv 0$ or $1\left(\bmod p^{k}\right)$.