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