Note that 143 = 11*13.

(a) If a is coprime with 143 explain why a¹²⁰ 1 mod 143.

(b) If a is coprime with 143, show that, in fact, a⁶⁰ 1 mod 143.

Hint. When u, v are coprime, we know that b c mod uv if and only b c mod u

and b c mod v. This is the essential fact behind the Chinese Remainder Theorem.