modular arithmetic - Proving identities (mod $pq$) using Fermat's little theorem?

I have come across this question, which reminded me of Fermats little theorem, i dont know if the Fermats theorem is actually in use in the following mathematical statements

an integer a is a coprime with p and a coprime with q (p and q are different prime numbers )then prove that

$$a^{(p-1)(q-1)} \equiv 1(\operatorname{mod} pq)$$

$$p^{q-1}+q^{p-1} \equiv 1(\operatorname{mod} pq)$$

any help would be appreciated, thanks in advance.