# Reduced Residue System/Examples

## 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 $5$

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 $18$.