Reduced Residue System Modulo Prime

Theorem
Let $p$ be a prime number.

The reduced residue system modulo $p$ contains $p - 1$ elements:


 * $Z'_p = \left\{ {\left[\!\left[{1}\right]\!\right]_m, \left[\!\left[{2}\right]\!\right]_m, \ldots, \left[\!\left[{p - 1}\right]\!\right]_m}\right\}$

and so can be defined as:
 * $Z'_p = Z_p \setminus \left\{ {\left[\!\left[{0}\right]\!\right]_m}\right\}$

Proof
From Prime not Divisor implies Coprime, each of $1, 2, \ldots, p - 1$ is coprime to $p$.

The result follows by definition of reduced residue system modulo $p$.