Complete Residue System/Examples/Modulo 11/Even Integers
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Examples of Complete Residue Systems
- $\set {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}$
forms a complete residue system modulo $11$.
Proof
We have:
\(\ds 12\) | \(=\) | \(\ds 1 \times 11 + 1\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 14\) | \(=\) | \(\ds 1 \times 11 + 3\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 3\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 16\) | \(=\) | \(\ds 1 \times 11 + 5\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 18\) | \(=\) | \(\ds 1 \times 11 + 7\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 7\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 20\) | \(=\) | \(\ds 1 \times 11 + 9\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 9\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 22\) | \(=\) | \(\ds 2 \times 11 + 0\) | ||||||||||||
\(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {11}\) |
Thus we see that:
- $\set {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}$
is equivalent to:
- $\set {2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 0}$
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 $1 \ \text {(c)}$