This exercise investigates what happens if we drop the assumption that gcd(e, p − 1) = 1 in Proposition 3.2. So let p be a prime, let c ≡ 0 (mod p), let e ≥ 1, and consider the congruence xe ≡ c (mod p). (3.36) (a) Prove that if (3.36) has one solution, then it has exactly gcd(e, p − 1) distinct solutions. (Hint. Use primitive root theorem (Theorem 1.30), combined with the extended Euclidean algorithm (Theorem 1.11) or Exercise 1.27.) (b) For how many non-zero values of c (mod p) does the congruence (3.36) have a solution?