The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$

Question

I'm trying to find a proof of this:

The group $\langle\mathbb{Z}_n,\oplus\rangle$ is cyclic for every $n$, where $1$ is a generator. The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$, as the reader can prove as an exercise.

It is perfectly clear that $1$ generates all $\mathbb{Z}_n$, but I can't get myself to understand the second part or find a way to prove it. Thanks.

Answer 1

Hint:

Use Bézout's identity to prove that if $\,\gcd(g,n)=1$, $1$ can be obtained as a multiple of $g$ (in $\mathbf Z_n$).