2. (a) Let m = p1 . . . Pn be a product of distinct primes, and assume that god(k, (m) ) = 1 so that Ju with ku = 1 (mod p(m)). Prove that x* = b (mod m) has unique solution x = b" (mod m), regardless of whether god (b, m) = 1. (Hint: Carefully read the proof of Theorem 6.7) (b) Consider the congruence x = 6 (mod 9). Show that you can find u satisfying 5u = 1 (mod p(9) ), but that x = 6" is not a solution to the required congruence. Can you identify where the distinct prime condition was needed in part (a)? (c) Solve the congruence x =3 (mod 1155)