Residue Class

The residue classes of a function f(x) mod n are all possible values of the residue f(x) (mod n). For example, the residue classes of x^2 (mod 6) are {0,1,3,4}, since

 0^2=0 (mod 6)
1^2=1 (mod 6)
2^2=4 (mod 6)
3^2=3 (mod 6)
4^2=4 (mod 6)
5^2=1 (mod 6)

are all the possible residues.

A complete residue system is a set of integers containing one element from each class, so {0,1,9,16} would be a complete residue system for x^2 (mod 6).

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

See also

Complete Residue System, Congruence, Cubic Number, Quadratic Reciprocity Theorem, Quadratic Residue, Reduced Residue System, Residue, Square Number

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Residue Class

Cite this as:

Weisstein, Eric W. "Residue Class." From MathWorld--A Wolfram Web Resource.

Subject classifications