# Symbols:Z/Reduced Residue System

## Reduced Residue System Modulo $m$

$\Z'_m$

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 $m$.

The $\LaTeX$ code for $\Z'_m$ is \Z'_m  or \mathbb Z'_m or \Bbb Z'_m.