Show that $\mathbb{Z}_n$ always has an even number of idempotents.

Question

$a$ is an idempotent if $a^2=a$.

Show that $\mathbb{Z}_n$ always has an even number of idempotents.

Answer 1

Hint:

In any commutative ring, if $a$ is an idempotent, then $a'=1-a$ is an idempotent. These idempotents are orthogonal, i.e. $aa'=0$.

On the other hand, if $a'$ were equal to $a$, we would have $a^2=0$. This cannot be, as $a=a'\iff2a=1$, which implies $a$ is a unit.