In my lecture notes there is a lemma which states the following :
Suppose $a_1,....a_{\phi(n)}$ is a complete set of residues , coprime to $n$, if $k \in \Bbb N, s.t.(k,n)=1$, then $ka_1, ka_2, ….,ka_{\phi(n)}$ is also a complete set of residues coprime to $n$.
I'm pretty sure that it should be a reduced residue system rather than a complete residue system though right ? seems as how the euler phi function only counts numbers relatively prime to n.
Yes, you're right. This is definitely a reduced residue system.
Given a set $$S=\{1,2,\cdots,n\},$$ a complete residue system is a set $T=\{a_1,\cdots,a_n\}$ of $n$ integers incongruent $\pmod n$.
Conversely, a set $R=\{k \in S : (k,n)=1\}=\{b_1,\cdots,b_{\phi(n)}\}$ of incongruent elements is a reduced residue system and has $\phi(n)$ elements.