Finding the number of a roots in an equation modulo n

Question

I am looking for some n such that $$x^2+x=0\pmod{n}$$ has at least 4 solutions. Is there any way of doing this reasonably quickly without having to check every solution manually? There must be.

Unfortunately, I can't seem to figure out what that way is. I am rather new to fields and modular arithmetic. Maybe I simply have not yet learned enough?

Thanks

Answer 1

Let $n=\prod_{i=1}^m{p_i}^{r_i}$ where $p_i$s are prime.

So for each $p_i,1\le i\le m, n|x(x+1)\implies {p_i}^{r_i}|x(x+1)$

But $(x+1,x)=1\implies$ either ${p_i}^{r_i}|x$ or ${p_i}^{r_i}|(x+1)$ giving two solutions.

So using Chinese Remainder Theorem, we shall get $2^m$ solutions