Condition for invertible residue class

Question

During my studies of elementary number theory, I came across the following problem:

Let $n \in \mathbb{N}$ and $k' \in \mathbb{Z}/n\mathbb{Z}$. Show that $k'$ is invertible iff $n$ and $k$ are relatively prime.

Answer 1

If $k'$ is invertible, let $k\in k'$ and $u \in \Bbb Z$ such that $ku \equiv 1 [n]$, then there exists $v\in \Bbb Z$ such that $ku + nv = 1$ which is a Bézout identity that ensures that $k$ and $n$ are coprime.

A nearly identical proof gives you the reverse implication.