Definition:Reduced Residue System/Least Positive

Definition
Let $n \in \Z: n \ge 1$.

Let $\phi \left({n}\right)$ be the Euler phi function of $n$.

The reduced residue system modulo $n$ is a set of integers:
 * $\left\{{a_1, a_2, \ldots, a_{\phi \left({n}\right)}}\right\}$

with the following properties:
 * each of which is prime to $n$
 * no two of which are congruent modulo n.

Also known as a reduced set of residues modulo $n$.

Each of the residue classes in this system can be referred to as a relatively prime residue class or coprime residue class.

Least Positive Residues
If each element of $\left\{{a_1, a_2, \ldots, a_{\phi \left({n}\right)}}\right\}$ is a positive integer less than or equal to $n$, this is called the reduced set of least positive residues modulo $n$.

Examples
The reduced set of least positive residues modulo $n$ for the first few integers are: