Proof of U(n) being automorphism of $Z_n$.

Question

I want to prove that $r\in U(n), \alpha : Z_n \to Z_n, \alpha (s)=mod(rs, n)$ is an automorphism. I have proved that $\alpha$ is one-one, maps identity to identity and $\alpha(s+t) = \alpha(s) + \alpha(t)$.

I am not able to prove the onto part.

Answer 1

Recall that $r$ is a unit of $\mathbb{Z}_n$ if and only if $r$ is coprime with $n$. Thus, by Bezout's identity, we have $s$ and $t$ so that $sn + tr = 1$. Modulo $n$, this gives $tr \equiv 1 \pmod{n}$. Therefore, for any $k \in \mathbb{Z}_n$,

$$ \alpha(kt) = ktr \equiv k \pmod{n} . $$