Units of Z/nZ are the residues coprime to n

Statement

Let $n \geq 1$ and $a$ be integers. The residue class $[a]$ is a unit of the ring $\mathbb{Z}/n\mathbb{Z}$ if and only if $\gcd(a, n) = 1$.

Remarks

If $\gcd(a, n) = 1$, then Bézout's lemma gives integers $x$ and $y$ with $a x + n y = 1$. So $[a][x] = [1]$. If $[a][x] = [1]$, then $a x - 1$ is a multiple of $n$, so each common divisor of $a$ and $n$ divides $1$. The condition does not depend on the representative, because $a \equiv b \pmod{n}$ implies $\gcd(a, n) = \gcd(b, n)$. For $n = 12$, the units are $[1]$, $[5]$, $[7]$ and $[11]$.

Explanations

L1 human

A residue has a multiplicative inverse modulo $n$ exactly when it has no common factor with $n$. Bézout's lemma gives the inverse. So the group of units of Z/nZ is the set of residues coprime to $n$.