A number $a$ has a multiplicative inverse modulo $n$ when there is an integer $x$ such that

$$ax\equiv1\pmod n.$$

Such an inverse exists exactly when

$$\gcd(a,n)=1.$$

Bézout's identity explains why. If

$$ax+ny=1,$$

then reducing modulo $n$ gives

$$ax\equiv1\pmod n.$$

Linear congruences

A congruence

$$ax\equiv b\pmod n$$

can be solved by multiplying by the inverse of $a$ when that inverse exists.

Unlike ordinary arithmetic, division modulo $n$ is therefore not always valid: it depends on whether the divisor is invertible in the chosen modulus.

This distinction is fundamental in modular equations, cryptographic constructions and algorithms over finite residue classes.