Proof: Uniqueness of multiplicative inverse

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: imjm1(modn). By the definition of congruence modulo n, im=pn+1 for some integer p, yielding the Bézout’s identify 1=impn. Since 1 clearly divides m and n, gcd(m,n)=1 by the Bézout's lemma. Thus, ij1(modn) by the cancellation law in modular arithmetic. Q.E.D.