Mdoular arithmetic is essentially arithmetic on remainders. Recall that whenever we have two integers , with , there exist unique integers and , with such that The number is the remainder.
Doing arithmetic on remainders is usually much easier than doing arithmetic on the original numbers, especially if is small. In addition, since the set of remainders is finite for any given , modular arithmetic reduces a problem on the infinite set of integers to a finite set. This observation is often useful in solving number theoretic problems, as we will see in later posts.
While modular arithmetic can be introduced in a completely elementary fashion, I will introduce this by group theory. Specifically, let is a group and be a subgroup. Define as cosets of the form . Observe that if and only if . A subgroup is normal if for all , equals . It is not too difficult to show that is a group if and only if is a normal subgroup of . The group operation is simply .
So what does this have to do with modular arithmetic? In this case, and . Note that is abelian (i.e. for all ). Consequently, any subgroup including is normal since . Thus is a group. Each integer in is represented by its remainder when divided by . Modular arithmetic is nothing other than group operation on .