# Congruence class's proof of with 0 divisors

Question

Let n be a positive integer and consider $[a]$ in $Zn$. Prove that $[a]$ is a zero divisor in $Zn$ if and only if it does not have an inverse in $Zn$.

An element $[a]$ of $Zn$ has a multiplicative inverse in $Zn$ if and only if $(a,n)=1$ in $Zn$ where n is prime this will always be true and the theorem holds.