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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


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Bibliography (further reading)

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

[1272] Landau, Edmund: “Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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