# Reduced Residue System/Examples

## Examples of Reduced Residue Systems

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

 $\ds 1$ $:$ $\ds \set 1$ $\ds 2$ $:$ $\ds \set 1$ $\ds 3$ $:$ $\ds \set {1, 2}$ $\ds 4$ $:$ $\ds \set {1, 3}$ $\ds 5$ $:$ $\ds \set {1, 2, 3, 4}$ $\ds 6$ $:$ $\ds \set {1, 5}$ $\ds 7$ $:$ $\ds \set {1, 2, 3, 4, 5, 6}$ $\ds 8$ $:$ $\ds \set {1, 3, 5, 7}$ $\ds 9$ $:$ $\ds \set {1, 2, 4, 5, 7, 8}$ $\ds 10$ $:$ $\ds \set {1, 3, 7, 9}$

### 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 $5$

The set of integers:

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

#### Arithmetic Sequence

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

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

#### Square Numbers

The set of integers:

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

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