Show that if a and m are relatively prime positive integers, then the inverse of a modulo m is unique modulo m. [Hint: Assume that there are two solutions b and c of the congruence ax ? 1 (mod m). Use Theorem 7 of Section 4.3 to show that b ? c (mod m).]

Solution Step 1 We know that Inverse of a modulo m is an integer j such that Let there be two inverses of modulo m , b and c So, and By transitivity property of congruence, we get Since and Therefore, which is the unique inverse of a modulo m .