Reduced Residue System/Examples/Modulo 18/Arithmetic Sequence
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Examples of Reduced Residue Systems
The arithmetic sequence:
- $\set {5, 11, 17, 23, 29, 35}$
does not form a reduced residue system modulo $18$.
Proof
We have:
\(\ds 23\) | \(=\) | \(\ds 1 \times 18 + 5\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {18}\) | |||||||||||
\(\ds 29\) | \(=\) | \(\ds 1 \times 18 + 11\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 11\) | \(\ds \pmod {18}\) | |||||||||||
\(\ds 35\) | \(=\) | \(\ds 1 \times 18 + 17\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 17\) | \(\ds \pmod {18}\) |
Thus we see that:
- $\set {5, 11, 17, 23, 29, 35}$
is equivalent to:
- $\set {1, 5, 17}$
While $\set {1, 5, 17}$ 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 $2 \ \text {(b)}$