Residue Class

The residue classes of a function  mod  are all possible values of the residue . For example, the residue classes of  (mod 6) are , since

are all the possible residues.

A complete residue system is a set of integers containing one element from each class, so  would be a complete residue system for  (mod 6).

The  residue classes prime to  form a group under the binary multiplication operation (mod ), where  is the totient function (Shanks 1993) and the group is classed a modulo multiplication group.

REFERENCES:

Nagell, T. "Residue Classes and Residue Systems." §20 in Introduction to Number Theory. New York: Wiley, pp. 69-71, 1951.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 56 and 59-63, 1993.