Question about a proof in Junod's paper about the Blum-Blum-Shub generator

Question

I am reading Junod's paper about the Blum-Blum-Shub generator. There is one thing I just do not understand about his proof of Lemma 4. On page 17 he writes:

We conclude that $x=-x_0$ and from Lemma 1 we know that $x \ne x_0$

(So far I understand.)

, so they must have different parities (Why is this the case?), $n$ beeing odd, which is a contradiction.

Could you please explain that to me?

Answer 1

If $x_0$ is odd, then $-x_0 = n - x_0$ must be even (and versa-visa).