Proposition 16

Let $a$ be an integer $> 0$. The integers $r$ such that $0 \leqslant r < a$ form a system of representatives of the equivalence relation $x \equiv y \pmod{a}$ on $\mathbf{Z}$.

If $x$ is an integer $\geqslant 0$, there exist (Set Theory, III, § 5, no. 6) integers $q$ and $r$ such that $x = aq + r$ and $0 \leqslant r < a$ and $x \equiv r \pmod{a}$. If $x$ is an integer $\leqslant 0$, the integer $-x$ is $\geqslant 0$ and by the above there exists an integer $r$ such that $0 \leqslant r < a$ and $-x \equiv r \pmod{a}$. Writing $r' = 0$ if $r = 0$ and $r' = a - r$ if $r > 0$, then

and $0 \leqslant r' < a$. We now show that if $0 \leqslant r < r' < a$, then $r \not\equiv r' \pmod{a}$. Now $r' - r < na$ for $n \geqslant 1$ and $r' - r > na$ for $n \leqslant 0$, whence $r' - r \notin a\mathbf{Z}$.