Why $99x^2 \equiv 1 \mod 5 \implies (-1)x^2 \equiv 1 \mod 5$

Question

Lastly I have read a example of some exercise. There was this statement: $$99x^2 \equiv 1 \pmod 5\quad \implies\quad (-1)x^2 \equiv 1 \pmod 5$$ Can somebody explain that simple fact to me?

Answer 1

$$99x^2\equiv_5100x^2-x^2\equiv_50-x^2\equiv_5-x^2$$

Answer 2

Since we have $$99\equiv -1 \mod 5$$ this is the reason.

Answer 3

Because $99\equiv 4 \equiv -1 \mod 5$

Answer 4

$$99\equiv 4=(5-1)\equiv -1 \pmod{5}$$

:)