Modular multiplicative inverse

Remarks

An inverse of $a$ modulo $n$ exists if and only if $\gcd(a, n) = 1$. See existence of inverses modulo n. If $x$ is an inverse of $a$, then the inverses of $a$ are exactly the integers $y \equiv x \pmod n$. So the inverse is unique as a residue class, and we write it as $a^{-1} \bmod n$. In the ring of integers modulo n, the class $\overline{x}$ is the inverse of $\overline{a}$. For example, $3^{-1} \equiv 5 \pmod 7$, because $3 \cdot 5 = 15 \equiv 1 \pmod 7$.