Definition:Complete Residue System

Definition

Let $m \in \Z_{\ne 0}$ be a non-zero integer.

Let $S := \set {r_1, r_2, \dotsb, r_s}$ be a set of integers with the properties that:

$(1): \quad i \ne j \implies r_i \not \equiv r_j \pmod m$

$(2): \quad \forall n \in \Z: \exists r_i \in S: n \equiv r_i \pmod m$

Then $S$ is a complete residue system modulo $m$.

Examples

Complete Residue Systems Modulo $3$

The following sets of integers are complete residue systems modulo $3$:

$\set {1, 2, 3}$

$\set {0, 1, 2}$

$\set {-1, 0, 1}$

$\set {1, 5, 9}$

Complete Residue Systems Modulo $11$

Powers of $2$

The set of integers:

$\set {0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512}$

forms a complete residue system modulo $11$.

Odd Integers

The set of integers:

$\set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}$

forms a complete residue system modulo $11$.

Even Integers

The set of integers:

$\set {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}$

forms a complete residue system modulo $11$.

Least Absolute Residues

The set of integers:

$\set {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}$

forms a complete residue system modulo $11$.

Also see

• Definition:Residue (Modulo Arithmetic)

• Definition:Residue Class
• Definition:Set of Residue Classes

• Definition:Reduced Residue System

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Definition $\text {4-3}$