Reduced Residue System/Examples/Modulo 18/Least Positive Residues

Examples of Reduced Residue Systems

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

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

Proof

The least positive residues of $18$ are:

$S := \set {0, 1, 2, \dotsc, 17}$

We have that:

$18 = 2 \times 3^2$

so the least positive reduced residue system of $18$ is the set of elements of $S$ which have neither $2$ or $3$ as a prime factor.

That is:

$S \setminus \set {0, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16}$

Hence the result.

$\blacksquare$

Sources

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