I know that a reduced residue system consists of elements from a complete residue system (of, say, $mod(n)$) which satisfy:
- $(a,n) = 1$ for each $a$ in the complete residue system
- A reduced residue system has $φ(n)$ elements
However, I don't understand the significance of articulating this. In what way is it useful? What possible information can we derive from an element's reduced residue system?
Reducing it so that all elements are coprime to $n$ makes sure that all elements have an inverse (a multiplicative one, that is). Therefore, the system we work in is a (finite) group under $(\cdot)$, so we can do fun things with them like proving (as mentioned in the comments) that for each $a$ in that group we have
$$a^{\varphi(n)}=1$$
and for example, that the amount of generators in that group (provided there is one) is $\varphi(\varphi(n))$. In general, knowing that each element has a multiplicative inverse is very useful, and that is why we can do much more with those systems when $n$ is prime.