🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

With $n\in {\mathbb{N}}_1$ then the relation $aRb$ defined as $a\%n=b\%n$ induces an equivalence relation therefore given any $a\in \mathbb{Z}$ we get an equivalence class $\left[a\right]$, and we define $\bar{a}:=\left[a\right]$ to lighten the standard notation of equivalence classes

Suppose that $n\in {\mathbb{N}}_1$, then we define $${\displaystyle {\mathbb{Z}}_n^{\%}:=\{\bar{0},\bar{1},\bar{2},\dots ,\overline{n-1}\}}$$ and call this set the integers modulo $n$

Suppose that $a,b\in \mathbb{Z}$, then we define $\bar{a}+\bar{b}=\overline{a+b}$

Suppose that $a,b\in \mathbb{Z}$, then we define $\bar{a}·\bar{b}=\overline{a·b}$