(a) State and prove the Fermat-Euler Theorem. Deduce Fermat's Little Theorem. State Wilson's Theorem.

(b) Let $p$ be an odd prime. Prove that $X^{2} \equiv-1(\bmod p)$ is solvable if and only if $p \equiv 1(\bmod 4)$.

(c) Let $p$ be prime. If $h$ and $k$ are non-negative integers with $h+k=p-1$, prove that $h ! k !+(-1)^{h} \equiv 0(\bmod p) .$