Show equivalence for different $n$

Question

Let $\{ a_1, \dots, a_n \}$ be a complete set of residues modulo $n$. How can we show that

$a_1+ \dots+ a_n\equiv 0 \pmod{n}$ if $n$ is odd, and

$a_1+ \dots+ a_n\equiv \frac{n}{2} \pmod{n}$ if $n$ is even?

Could you give me a hint?

Answer 1

$$ 0+1+2+...+(n-1) =\frac {n(n-1)}{2} $$

Notice that $\frac {n(n-1)}{2}\equiv 0 \pmod{n}$ if $n$ is odd and $\frac {n(n-1)}{2}\equiv n/2 \pmod{n}$ even if $n$ is even.

Answer 2

A quick hint: for any $k$ such that $1 \leq k < n$ consider $(n - k) + k$.