Prove that if $$r_1,r_2,...,r_{\phi(m)}$$ is a reduced residue system modulo $m$, and $m$ is odd, then $$m|r_1+r_2+...+r_{\phi(m)}$$
My attempt at proof was to first prove that for any two natural numbers $a,b$, if $GCD(a,b)=1$, then $GCD(a-b,b)=1$. I then used this to state that if some $r_i$ is part of the reduced residue system, $m-r_i$ must be as well, so each $r_i$ in the sum can be paired with $m-r_i$ and the sum telescopes to $m+m+...+m$.
Is this correct? If so, what is the role of "and $m$ is odd" in the question?