Solve $x^2 -3x+2 \equiv 0 \pmod 6$

Question

I need help understading how to solve this equation. $x^2-3x+2 \equiv 0 \pmod 6 $

I'm having trouble with the said equation.

Answer 1

First notice that $x^2-3x+2=(x-1)(x-2)$

So we want any $x$ where any of the following are met: $$x\equiv 1 \pmod 6$$ $$x\equiv 2\pmod 6$$ $$x \equiv 1\pmod 2\text{ and } x\equiv 2\pmod 3$$ $$x \equiv 0\pmod 2\text{ and } x\equiv 1\pmod 3$$

See if you can deduce when these are.