Application of Fermat's Theorem?

Question

If p and q are distinct primes and gcd(a,p*q)=1 and
n=lcm(p-1,q-1), then show that a^n = 1(modulo p*q).

Okay, we have gcd(a,p)=gcd(a,q)=1 since p and q are distinct. Then by Fermat's Theorem,

a^(p-1)=1 (modulo p) and a^(q-1)=1 (modulo q). Then how can we proceed?

Answer 1

Given that n is a multiple of p-1 and q-1, a^n =1 mod p and mod q.

Well, if k = 1 mod b and mod c, and b and c are relatively prime, k = 1 mod bc. That's just an application of the Chinese Remainder Theorem.