**Definition**: Complete Residue System

Let $m > 0$ be a positive integer. A **complete residue system modulo $m$** $\operatorname{crn}(m)$ is a subset $C\subset\mathbb Z$ of exactly $m$ integers such that each element $a\in C$ corresponds to exactly one possible congruence class $a(m).$

In other words, $C$ consists of *some given* $m$ integers $a_1,\ldots,a_m$ *representing* the equivalence classes $a_1(m),\ldots,a_m(m)$ being the elements of the quotient set $\mathbb Z_m.$

### Examples

The following are complete residue systems modulo $m$:

$$\begin{array}{rcl}C_1&=&\{0,1,\ldots,m-1\}.\\

C_2&=&\{1,2,\ldots, m\},\\

C_3&=&\{\lfloor\frac{-m}{2}\rfloor,\lfloor\frac{-m}{2}\rfloor+1,\ldots, \lfloor\frac{m}{2}\rfloor\}.\\

\end{array}$$

| | | | | created: 2019-04-19 07:32:41 | modified: 2019-04-19 07:48:16 | by: *bookofproofs* | references: [1272], [8152]

## 1.**Lemma**: Coprimality and Congruence Classes

## 2.**Proposition**: Creation of Complete Residue Systems From Others

[8152] **Jones G., Jones M.**: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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