Congruence of integers
One of the most useful distinctions amongst the integers is that between even and odd numbers. We have already seen a number of mathematical arguments which make use of this. The distinction between these two types of number can be expressed in terms of the remainder when the number is divided by 2: the remainder is 0 in the case of an even number and 1 for an odd number. The idea of congruence is a generalization of this: we classify integers according to the remainder after division by some fixed positive integer, not necessarily 2.
To be more precise, suppose that m is a positive integer. Two integers a and b are ...