## Quick Reference

(modulo n)

An equivalence class for the equivalence relation of congruence modulo *n*. So, two integers are in the same class if they have the same remainder upon division by *n*. If [*a*] denotes the residue class modulo *n* containing *a*, the residue classes modulo *n* can be taken as [0], [1], [2],…, [*n*−1]. The sum and product of residue classes can be defined by

[*a*] + [*b*] = [*a* + *b*], [*a*][*b*] = [*ab*],

where it is necessary to show that the definitions here do not depend upon which representatives *a* and *b* are chosen for the two classes. With this addition and multiplication, the set, denoted by **Z**_{n}, of residue classes modulo *n* forms a ring (in fact, a commutative ring with identity). If *n* is composite, the ring **Z**_{n} has divisors of zero, but when *p* is prime **Z**_{p} is a field.

*Subjects:*
Mathematics.