When does a number have a multiplicative inverse modulo a nonprime modulus?

In number theory, an integer $a$ has a multiplicative inverse modulo a modulus $N$ if and only if $a$ and $N$ are relatively prime—meaning their greatest common divisor is 1.

When the modulus $N$ is not prime, division is not always possible for every residue class. Specifically, the equation $a\times X\equiv 1\left(\mathrm{mod}N\right)$ has a solution for $X$ if and only if $a$ and $N$ share no common factors other than 1. If $a$ and $N$ are not relatively prime, no such $X$ exists, and the multiplicative inverse is undefined for that class.1