# Reduced Residue System/Examples

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## Contents

## Examples of Reduced Residue Systems

The reduced sets of residues modulo $n$ for the first few (strictly) positive integers are:

\(\displaystyle 1\) | \(:\) | \(\displaystyle \set 1\) | |||||||||||

\(\displaystyle 2\) | \(:\) | \(\displaystyle \set 1\) | |||||||||||

\(\displaystyle 3\) | \(:\) | \(\displaystyle \set {1, 2}\) | |||||||||||

\(\displaystyle 4\) | \(:\) | \(\displaystyle \set {1, 3}\) | |||||||||||

\(\displaystyle 5\) | \(:\) | \(\displaystyle \set {1, 2, 3, 4}\) | |||||||||||

\(\displaystyle 6\) | \(:\) | \(\displaystyle \set {1, 5}\) | |||||||||||

\(\displaystyle 7\) | \(:\) | \(\displaystyle \set {1, 2, 3, 4, 5, 6}\) | |||||||||||

\(\displaystyle 8\) | \(:\) | \(\displaystyle \set {1, 3, 5, 7}\) | |||||||||||

\(\displaystyle 9\) | \(:\) | \(\displaystyle \set {1, 2, 4, 5, 7, 8}\) | |||||||||||

\(\displaystyle 10\) | \(:\) | \(\displaystyle \set {1, 3, 7, 9}\) |

### Modulo $18$

#### Least Positive Residues

The least positive reduced residue system of $18$ is the set of positive integers:

- $\set {1, 5, 7, 11, 13, 17}$

#### Powers of $5$

- $\set {1, 5, 25, 125, 625, 3125}$

forms a reduced residue system modulo $18$.

#### Arithmetic Progression

- $\set {5, 11, 17, 23, 29, 35}$

does not form a reduced residue system modulo $18$.

#### Square Numbers

- $\set {1, 25, 49, 121, 169, 289}$

does not form a reduced residue system modulo $18$.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {4-2}$ Residue Systems: Example $\text {4-8}$