Complete Residue System/Examples/Modulo 11/Odd Integers
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Examples of Complete Residue Systems
- $\set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}$
forms a complete residue system modulo $11$.
Proof
We have:
| \(\ds 11\) | \(=\) | \(\ds 1 \times 11 + 0\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {11}\) | |||||||||||
| \(\ds 13\) | \(=\) | \(\ds 1 \times 11 + 2\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod {11}\) | |||||||||||
| \(\ds 15\) | \(=\) | \(\ds 1 \times 11 + 4\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 4\) | \(\ds \pmod {11}\) | |||||||||||
| \(\ds 17\) | \(=\) | \(\ds 1 \times 11 + 6\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 6\) | \(\ds \pmod {11}\) | |||||||||||
| \(\ds 19\) | \(=\) | \(\ds 1 \times 11 + 8\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 8\) | \(\ds \pmod {11}\) | |||||||||||
| \(\ds 21\) | \(=\) | \(\ds 1 \times 11 + 101\) | ||||||||||||
| \(\ds \) | \(\equiv\) | \(\ds 10\) | \(\ds \pmod {11}\) |
Thus we see that:
- $\set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}$
is equivalent to:
- $\set {1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10}$
The result follows from Initial Segment of Natural Numbers forms Complete Residue System.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Exercise $1 \ \text {(b)}$