Congruence

In this section, we introduce a concept of fundamental importance that will revolutionize the way we regard problems concerning divisibility. Albeit the underlying ideas have Indian and Chinese origins and Euler investigated some basic properties of remainders, it was Gauss who, in 1801, introduced the modern concepts of congruence and the arithmetic of residue classes to European audiences in Disquisitiones arithmeticae (Arithmetical Investigations) when he was 24. Gauss considered number theory to be the queen of mathematics. To him, its magical charm and inexhaustible wealth of intriguing problems placed it on a level way above other branches of mathematics. We owe a debt of gratitude to mathematicians such as Euler, Lagrange, Legendre, and Gauss for treating number theory as a branch of mathematics and not just a collection of interesting problems.

Given three integers a, b, and m, with m ≥ 2, we say that a is congruent to b modulo m, denoted by a ≡ b (mod m), if a and b yield the same remainder or residue when divided by m. Equivalently, a ≡ b (mod m), if there is an integer k such that a − b = km, that is, their difference is divisible by m.