$m,n>1$ are relatively prime integers , then are there at-least four idempotent (w.r.t. multiplication) elements in $\mathbb Z_{mn}$ ?

Question

If $m,n>1$ are integers such that $g.c.d.(m,n)=1$ then is it true that there are at-least four elements in $\mathbb Z_{mn}$ such that $x^2=x$ ( i.e. idempotent ) ?

Answer 1

Using the Chinese remainder theorem, $\Bbb Z_{mn}$ has the same structure as $\Bbb Z_{m}\times \Bbb Z_{n}$.

In this algebraic structure, there are the idempotents: $$ (0,0), (0,1), (1,0), (1,1) $$