Complete Residue System/Examples/Modulo 11/Even Integers

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Examples of Complete Residue Systems

The set of integers:

$\set {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}$

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