Complete Residue System/Examples/Modulo 11/Least Absolute Residues

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Examples of Complete Residue Systems

The set of integers:

$\set {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}$

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