Past exam of the options I course of the University of Oxford / 2012 / C / C1 / b / Solution

Let $R_n=\{k\in \mathbb{Z}:0<k<n,\gcd (k,n)=1\}$. The map $k\mapsto n-k$ sends $R_n$ to itself becauseApplying it twice returns $k$, so it is a bijection. Reindexing the finite sum by this bijection givesThere is no need to assume that every residue belongs to a pair of distinct elements; the reindexing also handles the fixed point when $n=2$.