residue class

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(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 Zn, of residue classes modulo n forms a ring (in fact, a commutative ring with identity). If n is composite, the ring Zn has divisors of zero, but when p is prime Zp is a field.

Subjects: Mathematics.

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