Notes On Number Theory And

ements. We will see in a moment that it is possible to do some arithmetic with these elements. Think of the integers as being written on a long ribbon, which is then wrapped around a circle of circumference m. The numbers that fall on top of (or underneath) 0 constitute the congruence class [0] m , and similarly for the congruence classes of 1, 2, . . . , m-1. Each of the m congruence classes contains infinitely many elements. Any two elements in the same congruence class di#er by a multiple of m; and elements of di#erent congruence classes always di#er by an amount that is not a multiple of m. The congruence classes are also called residue classes, since they concern the remainders, or residues, left after division by m. To work with , we can deal either with entire congruence classes or else just pick convenient representatives from each congruence class. A set of integers that contains exactly one member of each congruence class in is called a complete set of representative