Cyclicity Condition for Units of Ring of Integers Modulo m

Theorem
Let $n \in \Z_{\geq 0}$ be an integer.

Let $\left({\Z/n\Z,+,\cdot}\right)$ be the ring of integers modulo $n$.

Let $U = \left({ \left({ \Z/n\Z }\right)^\times,\cdot}\right)$ denote the group of units of this ring.

Then $U$ is cyclic if and only if $n = p^\alpha$ or $n = 2p^\alpha$, where $p \geq 3$ is prime and $\alpha \geq 0$.