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:

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

forms a reduced residue system modulo $18$.

Arithmetic Sequence

The 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 $18$.

Sources

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