Write the expression as a product of polynomials:

a (p-q) + q-p

Solution: Since p 6 = q are prime numbers, we have gcd (p, q) = 1. By Fermat’s Little Theorem, p q-1 ≡ 1 (mod q). Clearly q p-1 ≡ 0 (mod q). Thus p q-1 + q p-1 ≡ 1 (mod q). Exchanging the roles of p and q in the above argument, we prove that p q-1 + q p-1 ≡ 1 (mod p)