# Proof: Uniqueness of multiplicative inverse

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: $im\equiv jm\equiv 1\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n\right)$. 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, $gcd\left(m,n\right)=1$ by the Bézout's lemma. Thus, $i\equiv j\equiv 1\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n\right)$ by the cancellation law in modular arithmetic. Q.E.D.