Finding all the generators of $Z_{n}$

Question

I am to find all the generators of the cyclic group $Z_{n}$.
I know that $Z_{n}$ is a cyclic group because it can be generated by $\{1 \}$.
However I think that $Z_{n}$ can be generated not only by $\{1\}$ but by $\{1, 2, 3,4,...,n-1\}$. Unfortunately I don't know how can give the proof of that.

Answer 1

Your assertion is only true when $n$ is prime. For example, for $\mathbb{Z}_4$, $2_4$ is not a generator. So, it is only true if all the elements of $\mathbb{Z}_n$ have order exactly $n$, no more, no less.