Reduced Residue System/Examples/Modulo 18/Square Numbers
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Examples of Reduced Residue Systems
- $\set {1, 25, 49, 121, 169, 289}$
does not form a reduced residue system modulo $18$.
Proof
We have:
| \(\ds 25\) | \(=\) | \(\ds 1 \times 18 + 7\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 7\) | \(\ds \pmod {18}\) | |||||||||||
| \(\ds 49\) | \(=\) | \(\ds 2 \times 18 + 13\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 13\) | \(\ds \pmod {18}\) | |||||||||||
| \(\ds 121\) | \(=\) | \(\ds 6 \times 18 + 13\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 13\) | \(\ds \pmod {18}\) | |||||||||||
| \(\ds 169\) | \(=\) | \(\ds 9 \times 18 + 7\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 7\) | \(\ds \pmod {18}\) | |||||||||||
| \(\ds 289\) | \(=\) | \(\ds 16 \times 18 + 1\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {18}\) |
Thus we see that:
- $\set {1, 25, 49, 121, 169, 289}$
is equivalent to:
- $\set {1, 7, 13}$
While $\set {1, 7, 13}$ are all coprime to $18$, they do not form a least positive reduced residue system modulo $18$, which is:
- $\set {1, 5, 7, 11, 13, 17}$
from Least Positive Reduced Residue System Modulo 18.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Exercise $2 \ \text {(c)}$