# Definition: Complete Residue System

Let $m > 0$ be a positive integer. A complete residue system modulo $m$ 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 $m$:

$$\begin{array}{rcl}C_1&=&\{0,1,\ldots,m-1\}.\\ C_2&=&\{1,2,\ldots, m\},\\ C_3&=&\{\lfloor\frac{-m}{2}\rfloor,\lfloor\frac{-m}{2}\rfloor+1,\ldots, \lfloor\frac{m}{2}\rfloor\}.\\ \end{array}$$

Definitions: 1
Proofs: 2 3 4
Propositions: 5 6 7 8

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### References

#### Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998