Congruence classes and the arithmetic of remainders

Abstract

Throughout this chapter m will denote a fixed positive integer . 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 of integers. This is a typical process of mathematical abstraction: the set of all even integers is a more abstract notion than an even integer. In principle everything that is said about congruence classes can be reformulated in terms of congruent integers. However, as so often in mathematics the process of abstraction is enormously powerful because of the way in which it liberates our thinking. In this case one benefit of working with congruence classes is that they are finite in number so that we can make use of counting arguments like the pigeonhole principle introduced in Chapter 11. The power of this approach will be illustrated by giving an alternative proof of Theorem 20.1.5 which demonstrates that from an appropriate point of view the result is essentially ‘obvious’. A further application with far-reaching results known as Fermat's little theorem will be described in Chapter 24.