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)