Reduced Residue System Modulo Prime

Theorem

Let $p$ be a prime number.

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

$Z'_p = \set {\eqclass 1 m, \eqclass 2 m, \ldots, \eqclass {p - 1} m}$

and so can be defined as:

$Z'_p = Z_p \setminus \set {\eqclass 0 m}$

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 $p$.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Example $\text {4-9}$