Primitive Root is Generator of Reduced Residue System

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

Then:
 * $$\left\{{a, a^2, a^3, \ldots, a^{\phi \left({n}\right)}}\right\}$$

where $$\phi \left({n}\right)$$ is the Euler phi function of $$n$$, is a reduced residue system of $$n$$.

Thus the first $$\phi \left({n}\right)$$ powers of $$a$$ "generates" $$R$$.

We say that $$a$$ is a generator of $$R$$.

Proof
Let $$R = \left\{{a, a^2, a^3, \ldots, a^{\phi \left({n}\right)}}\right\}$$.

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

Suppose $$a^r \equiv a^s \left({\bmod\, n}\right)$$ where $$1 \le r \le s \le \phi \left({n}\right)$$.

Then $$a^{r-s} \equiv 1 \left({\bmod\, n}\right)$$.

From the definition of primitive root, the order of $a$ modulo $n$ is $$\phi \left({n}\right)$$.

So from Integer to Power of Multiple of Order $$\phi \left({n}\right)$$ divides $$r - s$$ and so $$r = s$$.

So no two elements are congruent modulo $n$.

So as $$R$$ contains $$\phi \left({n}\right)$$ integers none of which are congruent modulo $n$ to any of the others, $$R$$ is a reduced residue system of $$n$$.