Reduced Residue System/Examples/Modulo 18/Arithmetic Sequence

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Examples of Reduced Residue Systems

The arithmetic sequence:

$\set {5, 11, 17, 23, 29, 35}$

does not form a reduced residue system modulo $18$.


Proof

We have:

\(\ds 23\) \(=\) \(\ds 1 \times 18 + 5\)
\(\ds \) \(\equiv\) \(\ds 5\) \(\ds \pmod {18}\)
\(\ds 29\) \(=\) \(\ds 1 \times 18 + 11\)
\(\ds \) \(\equiv\) \(\ds 11\) \(\ds \pmod {18}\)
\(\ds 35\) \(=\) \(\ds 1 \times 18 + 17\)
\(\ds \) \(\equiv\) \(\ds 17\) \(\ds \pmod {18}\)

Thus we see that:

$\set {5, 11, 17, 23, 29, 35}$

is equivalent to:

$\set {1, 5, 17}$


While $\set {1, 5, 17}$ are all coprime to $18$, they do not form a least positive reduced residue system modulo $18$, which is:

$\set {1, 5, 7, 11, 13, 17}$

from Least Positive Reduced Residue System Modulo 18.

$\blacksquare$


Sources