Order of Group of Units of Integers Modulo m

Theorem
Let $n \in \Z_{\ge 0}$ 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:
 * $\left\vert{U}\right\vert = \phi \left({n}\right)$

where $\phi$ denotes the Euler $\phi$-function.

Proof
By Reduced Residue System under Multiplication forms Abelian Group, $U$ is equal to the set of integers modulo $n$ which are coprime to $n$.

It follows by Cardinality of Reduced Residue System:
 * $\left\vert{U}\right\vert = \phi \left({n}\right)$