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.   

Hint $\ $ By lil Fermat, both satisfy $\,x\equiv 1\,$ mod $\,p\ \&\ q\,$ thus $\,x\equiv 1\pmod{pq}\ $ by CCRT. Or, more elementarily, we can prove  it directly:  $\ p,q\mid x\!-1\!\,$ $\,\Rightarrow\,$ $\, pq\mid x\!-\!1,\,$ by $\,p,q\,$ coprime, by Euclid's Lemma or unique factorization. In fact the uniqueness of the CRT solution is equivalent to the universal property of lcm, i.e. $\ p,q\mid n\iff {\rm lcm}(p,q)\mid n$
