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 $m$:


 * $\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$

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 $m$.

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 $m$.

Some sources denote it $\Z^*_m$ or $Z^*_m$.