Reduced Residue System/Examples
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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 $5$
- $\set {1, 5, 25, 125, 625, 3125}$
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
- $\set {1, 25, 49, 121, 169, 289}$
does not form a reduced residue system modulo $18$.
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Example $\text {4-8}$