Modular Arithmetic

$({\mathbb{Z}}_m,+,\cdot )$ is a commutative ring:

1. Definition

$$\begin{array}{r}a\equiv b(\mathrm{mod}m)⟺a\mathrm{mod}m=b\mathrm{mod}m\end{array}$$

2. If $a\equiv b(\mathrm{mod}m)$ and $c\equiv d(\mathrm{mod}m)$, then

$$a+c\equiv b+d(\mathrm{mod}m),ac\equiv bd(\mathrm{mod}m)$$

$$\begin{array}{rl}(a+b)\mathrm{mod}m& =(a\mathrm{mod}m+b\mathrm{mod}m)\mathrm{mod}m\\ ab\mathrm{mod}m& =\left[(a\mathrm{mod}m)(b\mathrm{mod}m)\right]\mathrm{mod}m\end{array}$$