(related to Proposition: Creation of Complete Residue Systems From Others)

- By hypothesis, $m > 0$ is a positive integer, $n\perp m$ are co-prime, and $C=\{a_1,\ldots,a_m\}$ is a complete residue system modulo $m$.
- Since $C$ is complete, $a_i(m)\not\equiv a_j(m)$ if and only if $i\neq j.$
- For any given pair of indices $i,j$ with $i\neq j$ and by contraposition to the cancellation of congruences with factor co-prime to module, we have that $a_i(m)\not\equiv a_j(m)\Longrightarrow (a_in)(m)\not \equiv (a_jn)(m)$, provided $n\perp m.$
- Because any representative of $a(m)$ is either co-prime or not co-prime to $m$ we have $(a_in)(m)\not \equiv (a_jn)(m)$ if and only if $i\neq j.$
- That means that $nC:=\{na_1,\ldots,na_m\}$ is a complete residue system modulo $m.$∎