Modular multiplicative inverse

Definition

Let $n$ be a positive integer and let $a$ be an integer. A modular multiplicative inverse of $a$ modulo $n$ is an integer $x$ such that $a x \equiv 1 \pmod n$.

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 $3 \cdot 5 = 15 \equiv 1 \pmod 7$.

Explanations

L1 human

Modulo $n$, the inverse of $a$ plays the part of the fraction $1/a$. To solve $3 x \equiv 4 \pmod 7$, multiply both sides by $5$, the inverse of $3$. You get $x \equiv 20 \equiv 6 \pmod 7$.