Linear Equations Modulo m

Abstract

This chapter presents some fundamental ideas and methods of arithmetic. These include the greatest common divisor of two numbers, Euclid’s algorithm to find the greatest common divisor, and the Extended Euclidean Algorithm to write the greatest common divisor of two numbers as an integer linear combination of the two numbers (Bezout’s Identity). We show how to solve linear congruences and find all integer solutions of integer linear equations in two variables. A short section is devoted to what we call the “Coprime Divisibility Lemma”: if a number divides the product of two numbers and is coprime to one of them, it divides the other. That result and all of the methods of this chapter are basic tools for studying applications later in the book.