Definition:Reduced Residue System

Definition
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 $m$:


 * $\Z'_m = \left\{{\left[\!\left[{k}\right]\!\right]_m \in \Z_m: k \perp m}\right\}$

Thus $\Z'_m$ is the set of all coprime residue classes modulo $m$:
 * $\Z'_m = \left\{ {\left[\!\left[{a_1}\right]\!\right]_m, \left[\!\left[{a_2}\right]\!\right]_m, \ldots, \left[\!\left[{a_{\phi \left({m}\right)} }\right]\!\right]_m}\right\}$

where:
 * $\forall k: a_k \perp m$
 * $\phi \left({m}\right)$ denotes the Euler phi function of $m$.

Also known as
A reduced residue system modulo $m$ is also known as a reduced set of residues modulo $m$.

Some authors refer to this as the set of relatively prime residue classes modulo $m$.