Chapter 3.4, Problem 17E

let $U_n$ be the group of units as described in Exercise16. Prove that $\left[a\right]\in U_n$ if and only if $a$ and $n$ are relatively prime.

Exercise16

For an integer $n>1$, let $G=U_n$, the group of units in ${\mathbb{Z}}_n$ –that is, the set of all $\left[a\right]$ in ${\mathbb{Z}}_n$ that have multiplicative inverses. Prove that $U_n$ is a group with respect to multiplication.