Modulo operator properties

The modulo operator has the following properties:

• Reflexivity: $a{=}_{n}a$
• Symmetry: if $a{=}_{n}b$ then $b{=}_{n}a$
• Transitivity: if $a{=}_{n}b$ and $b{=}_{n}c$ then $a{=}_{n}c$
• $(a\bullet b){=}_n(a\text{ mod }n)\bullet (b\text{ mod }n)$ for $\bullet \in \{+,-,\times \}$

  $(a\bullet b)\text{ mod }n=(a\text{ mod }n)\bullet (b\text{ mod }n)\text{ mod }n$ $\begin{array}{rl}2& =(5\times 6)\text{ mod }4\\ & =[(5\text{ mod }4)\times (6\text{ mod }4)]\text{ mod }4\\ & =(1\times 2)\text{ mod }4\\ & =2\text{ mod }4\\ & =2\end{array}$

• if $a\times b{=}_{n}a\times c$ and $a$ is relatively prime to $n$, then $b{=}_{n}c$.

  $8\times 4{=}_{3}8\times 1$ $8$ is relatively prime to $3$ so $4{=}_{3}1$

• if $a_1{=}_n{b}_1$ and $a_2{=}_n{b}_2$ then $\begin{array}{rl}(a_1+a_2)& {=}_n(b_1+b_2)\\ & \text{and}\\ (a_1\times a_2)& {=}_n(b_1\times b_2)\end{array}$