Complete residue system

Definition

Let $n$ be a positive integer. A complete residue system modulo $n$ is a set $S$ of integers that contains exactly one element of each residue class modulo $n$.

Remarks

Equivalently, $S$ has exactly $n$ elements, and no two distinct elements of $S$ are congruent modulo $n$. The set $\{0, 1, \dots, n - 1\}$ is the set of the least nonnegative residues. It is a complete residue system modulo $n$. For odd $n$, the set $\{ -\tfrac{n-1}{2}, \dots, -1, 0, 1, \dots, \tfrac{n-1}{2} \}$ of the least absolute residues is also a complete residue system. If $S$ is a complete residue system modulo $n$ and $\gcd(a, n) = 1$, then $\{ a s + b : s \in S \}$ is also a complete residue system for each integer $b$.

Explanations

L1 human

A complete residue system picks one representative from each box of the integers modulo $n$. For $n = 4$, both $\{0, 1, 2, 3\}$ and $\{4, -3, 10, 7\}$ are complete residue systems.