What is meant by "Every number in a reduced residue system $\mod n$ is a generator for the additive group of integers modulo $n$"?

Question

What is meant by "Every number in a reduced residue system $\mod n$ is a generator for the additive group of integers modulo $n$"? I've no experience with Groups and Abstract algebra.

Answer 1

The meaning is that any of the reduced residue system elements $r$ mod $n$ will be a generator of the cyclic group $\mathbb{Z}/n\mathbb{Z}$. In other words every group element of $\mathbb{Z}/n\mathbb{Z}$ is expressed as an integer multiple $kr\bmod n$.

More explicitly the set $\{r,2r,\ldots,nr\}$ contains one representative of every residue class mod $n$.

This is because these "reduced residues" $r$ are coprime to $n$, as explained by the Wikipedia article you cited in a clarifying Comment:

Any subset $R$ of the integers is called a reduced residue system modulo $n$ if:

1. $\gcd(r, n) = 1$ for each $r$ contained in $R$;
2. $R$ contains $\varphi(n)$ elements;
3. no two elements of $R$ are congruent modulo $n$.

That is, since $r$ and $n$ are coprime, one has for suitable integers $a,b$ that:

$$ ar + bn = 1 $$

Then any residue $k \bmod n$ can be expressed as $kar \bmod n$.