Congruence classes and the arithmetic of remainders
The reader will be familiar with the idea that the set of integers is the disjoint union of two subsets: the set of even integers and the set of odd integers. In this case two integers lie in the same subset if and only if they are congruent modulo 2. In the same way, given any positive integer m we can divide the set of integers into m disjoint subsets, called the congruence classes modulo m: two integers lie in the same class if and only if they are congruent modulo m.
For some purposes, rather than using congruence of integers it is more convenient to use equality of congruence classes ...