The Relation of Congruence

Abstract

In this chapter, we define and explore the most basic properties of the important relation of congruence modulo n > 1. Our central goal is to prove the famous Fermat’s little theorem, as well as its generalization, due to Euler. The pervasiveness of these two results in elementary Number Theory owes a great deal to the fact that they form the starting point for a systematic study of the behavior of the remainders of powers of a natural number a upon division by a given natural number n > 1, relatively prime with a. We also present the no less famous Chinese remainder theorem, along with some interesting applications.