Modular Multiplicative Inverse

Inverses in ${\mathbb{Z}}_m$ (INV ${\mathbb{Z}}_m$)

Let $a$ be an integer with $1\le a\le m-1$. The element $[a]$ in ${\mathbb{Z}}_m$ has a multiplicative inverse if and only if $\gcd (a,m)=1$. Moreover, when $\gcd (a,m)=1$, the multiplicative inverse is unique.

Inverses in ${\mathbb{Z}}_p$ (INV ${\mathbb{Z}}_p$)

For all prime numbers $p$ and non-zero elements [a] in ${\mathbb{Z}}_p$, the multiplicative inverse $[a{]}^{-1}$ exists and is unique.