Cardinality of Reduced Residue System

Theorem
Let $n \ge 2$.

Let $\Z'_n$ be the reduced residue system modulo $n$.

Then:


 * $\left\vert{\Z'_n}\right\vert = \phi \left({n}\right)$

where $\phi \left({n}\right)$ is the Euler phi function.

Proof
Recall the definition of $\Z'_n$:


 * $\Z'_n = \left\{{\left[\!\left[{k}\right]\!\right]_n \in \Z_n: k \perp n}\right\}$

and the definition of $\phi \left({n}\right)$:


 * $\phi \left({n}\right) = \left\vert{\left\{{k: 1 \le k \le n, k \perp n}\right\}}\right\vert$

The result follows from Integer is Congruent to Integer less than Modulus.