2. Determine $x \in \mathbb{Z}$ for which $2|x-\sqrt{5}|+|x-2 \sqrt{5}| \in \mathbb{N}$.
2. If $x<\sqrt{5}$, it results in $2(\sqrt{5}-x)+2 \sqrt{5}-x=4 \sqrt{5}-3 x$, which does not verify. $.2 p$

If $\sqrt{5} \leq x<2 \sqrt{5}$, it results in $2(x-\sqrt{5})+2 \sqrt{5}-x=x$ and $x \in \mathbb{N}$. Since $\sqrt{5} \leq x<2 \sqrt{5}$, it follows that $x \in\{3,4\}$ $.3 p$

If $x \geq 2 \sqrt{5}$, it results in $2(x-\sqrt{5})+x-2 \sqrt{5}=3 x-4 \sqrt{5}$, which does not verify. $.2 p$