Let a, b be relatively prime positive integers, and b=[ (a), (b)] be the least common multiple of (a) and (b).

1. Prove that n^k 1(mod ab) for all integers n such that (n, ab) =1.

2. Assuming in additoin that a = p^rand b = q^swhere p and q are distinct odd primes and r, s > 1, explain why no smaller

exponentg

(i.e.,1g<k,withk=[(pr),(qs)]willresultinng1(modab)forallnsuchthat(n,ab)=1)