I've been wondering if there are proofs for these two questions... Searching has given me nothing. These aren't new results so they SHOULD be out there, but I can't find them.

1. Let p and q be distinct primes. Prove that, for any $\displaystyle a \in \mathbb{Z}$

$\displaystyle a^{pq} \equiv a^p + a^q (mod \ pq)$

2. Let $\displaystyle a < m$ be two positive integers. Prove that a and m are relatively prime if and only if there is some power of a which is an inverse of a modulo m.