Order of Group of Units of Integers Modulo m

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

Let $\struct {\Z / n \Z, +, \cdot}$ be the ring of integers modulo $n$.

Let $U = \struct {\paren {\Z / n \Z}^\times, \cdot}$ denote the group of units of this ring.


$\order U = \map \phi n$

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


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:

$\order U = \map \phi n$