Prove that any multiplicative inverse i of m modulo n is unique modulo n.
Let i and j be two multiplicative inverses of m modulo n: . By the definition of congruence modulo n, for some integer p, yielding the Bézout’s identify . Since 1 clearly divides m and n, by the Bézout's lemma. Thus, by the cancellation law in modular arithmetic. Q.E.D.