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The map $j\mapsto n-j$ is a bijection of the integers between one and $n-1$ that are coprime to $n$. If their sum is $S$, this bijection gives $S=n\phi (n)-S$, so $S=n\phi (n)/2$. This works even at $n=2$, where the single residue is fixed. The same identity follows from the weighted inclusion-exclusion principle by excluding multiples of the prime factors of $n$.