Z6n

Z6n denotes the ring of integers modulo 6n, typically written Z_{6n} or Z/6nZ. It consists of the set {0,1,...,6n−1} with addition and multiplication performed modulo 6n. For a fixed positive integer n, this is a finite commutative ring that is not a field when 6n is composite, and it has zero divisors except in trivial cases.

The additive group of Z/6nZ is cyclic of order 6n, and the ring's unit group U(Z/6nZ) comprises the residues that are coprime to 6n; its order is Euler's totient φ(6n).

By the Chinese Remainder Theorem, Z/6nZ is isomorphic to Z/2Z × Z/3Z × Z/nZ whenever n is coprime to 6.

In mathematics, Z/6nZ is used in modular arithmetic, number theory, and coding theory; it provides a compact setting for experiments with congruences, residues, and ring homomorphisms.