Prove that any multiplicative inverse i of m modulo n is unique modulo n.

Proof

Let i and j be two multiplicative inverses of m modulo n: $$im\equiv jm\equiv 1(\mathrm{mod}n)$$. By the definition of congruence modulo n, $$im=pn+1$$ for some integer p, yielding the Bézout’s identify $$1=im-pn$$. Since 1 clearly divides m and n, $$gcd(m,n)=1$$ by the Bézout's lemma. Thus, $$i\equiv j\equiv 1(\mathrm{mod}n)$$ by the cancellation law in modular arithmetic. Q.E.D.