Primitive Root is Generator of Reduced Residue System

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Theorem

Let $a$ be a primitive root of $n$.

Then:

$\set {a, a^2, a^3, \ldots, a^{\map \phi n} }$

where $\map \phi n$ is the Euler phi function of $n$, is the reduced residue system of $n$.

Thus the first $\map \phi n$ powers of $a$ "generates" the reduced residue system of $n$.

We say that $a$ is a generator of the reduced residue system of $n$.

Proof

Let $R = \set {a, a^2, a^3, \ldots, a^{\map \phi n} }$.

Each element of $R$ is coprime to $n$ as $a \perp n$.

Suppose $a^r \equiv a^s \pmod n$ where $1 \le r \le s \le \map \phi n$.

Then $a^{r - s} \equiv 1 \pmod n$.

From the definition of primitive root, the multiplicative order of $a$ modulo $n$ is $\map \phi n$.

So from Integer to Power of Multiple of Order $\map \phi n$ divides $r - s$ and so $r = s$.

So no two elements are congruent modulo $n$.

So as $R$ contains $\map \phi n$ integers none of which are congruent modulo $n$ to any of the others, $R$ is a reduced residue system of $n$.

$\blacksquare$