Modular Arithmetic

Definition

$n\in \mathbb{N}$ ${\mathbb{Z}}_n^{\times }:=\{[k]\mid (k,n)=1\}\subset {\mathbb{Z}}_n$

Proposition

$({\mathbb{Z}}_n^{\times },\cdot )$ is a group.

Euler $\phi$-function

$\phi (n)=|\{1\le k\le n\mid (n,k)=1\}|=|{\mathbb{Z}}_n^{\times }|.$