\begin{definition}[Definition:Reduced Residue System]
Let $m \in \Z_{> 0}$ be a (strictly) positive integer.
The '''reduced residue system modulo $m$''', denoted $\Z'_m$, is the set of all residue classes of $k$ (modulo $m$) which are prime to $m$:
:$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$
Thus $\Z'_m$ is the '''set of all coprime residue classes modulo $m$''':
:$\Z'_m = \set {\eqclass {a_1} m, \eqclass {a_2} m, \ldots, \eqclass {a_{\map \phi m} } m}$
where:
:$\forall k: a_k \perp m$
:$\map \phi m$ denotes the Euler phi function of $m$.
\end{definition}