CWMO 2010 Problem 1

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that**(a)** $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$ ;(b) $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$ .