System modular equation. Question.

Question

$$2x \equiv 4 \mod 8 \iff x \equiv 2 \mod 4 $$ And this is true, but is it a true?: $$\begin{cases} 2x \equiv 4 \mod 8 \\ x \equiv 2 \mod 6 \end{cases} $$ $$\iff$$ \begin{cases} x \equiv 2 \mod 4\\ x \equiv 2 \mod 6 \end{cases} $$ Why? Why not?

Answer 1

Check the following equivalences$$2x\equiv 4 \pmod 8 \iff 8 \mid (2x-4) \iff 4\mid (x-2) \iff x\equiv 2 \pmod 4 $$