ROUGH DRAFT authorea.com/104023

# Modular Arithmetic

Modular arithmetic is used in discrete math to find remainders. The definition states, if $$a$$ and $$b$$ are both integers and $$m>0$$  then $$a$$ is congruent to $$b$$ (mod $$m$$) if $$m$$ divides $$a-b$$. The notion of modular arithmetic deals with the remainders that are found in Euclidean division. The actions of trying to find the remainder is also known as modulo operation or (mod $$n$$) where $$n$$  is a an integer. For instance, the division of $$8$$ by $$3$$ can also be written as $$8$$ (mod $$3$$) and we can find the remainder to equal $$2$$ thus, $$8$$ (mod $$3$$) $$=2$$.
Properties:
1.) $$a\equiv b$$ (mod $$m$$) says that $$a$$ is congruent to $$b$$ ﻿(mod $$m$$).
2.) Both integers $$a$$ and $$b$$ are congruent if and only if they have the same remainders when dividing by $$m$$
3.) If $$a$$ $$\neg\equiv$$ $$b$$ (mod $$m$$) we can write $$a$$ not congruent to $$b$$ (mod $$m$$).
The addition rule in modular arithmetic shows that when $$a,b,c,$$ and $$d$$ are integers and $$m$$ is also a positive integer then,$$a+b\equiv c+d$$ (mod $$m$$) , where $$a\equiv c$$ (mod $$m$$) and $$b\equiv d$$ (mod $$m$$). The subtraction rule is similar,  $$a-c\equiv b-d$$ (mod $$m$$).