Exercise · ZW-COMPLETE-RESIDUE-SYSTEM-010 · ★★★ Advanced
Let \(m=6\). What is the result of multiplying the system \(\{0,1,2,3,4,5\}\) by \(a=5\) and reducing modulo \(6\)?
Choose one option.
A \(\{0,2,4,0,2,4\}\), because each multiple of \(5\) repeats B \(\{0,5,10,15,20,25\}\), which is not reduced C \(\{0,1,2,3,4,5\}\), because multiplication preserves order D \(\{0,5,4,3,2,1\}\), which is the same set of classes Show answer D. \(\{0,5,4,3,2,1\}\), which is the same set of classes← Back to the question
Explanation Since \(5k\equiv -k \pmod 6\), the remainders are \(0,5,4,3,2,1\), which are all the classes. Assuming that order is preserved is a common mistake, and the unreduced option is not a set of residues .
A complete residue system modulo \(m\) is a set of \(m\) integers that contains exactly one representative from each congruence class modulo \(m\).
\[\{r_1,\dots,r_m\}\ \text{with}\ r_i\not\equiv r_j\ (\bmod\ m)\ \forall i\neq j\]
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