Problem 2. Determine the non-zero natural number $x$, such that $\frac{\sqrt{3}+2 \sqrt{x}}{3 \sqrt{3}-\sqrt{x}} \in \mathbb{Z}$.
## Solution. 1p by default

$\frac{\sqrt{3}+2 \sqrt{x}}{3 \sqrt{3}-\sqrt{x}}=a$, where $a \in \mathbb{Z} \mathbf{(1 p)}$. It follows that $\sqrt{\frac{3}{x}}=\frac{2+a}{3 a-1} \in \mathbb{Q}$, hence $x=3 p^{2}, p \in \mathbb{N}^{*}(\mathbf{1 p})$

We obtain $a=\frac{2 p+1}{3-p} \in \mathbb{Z}(\mathbf{1 p}) \rightarrow 3-p \mid 2 p+1(\mathbf{0.5 p})$ and since $3-p \mid 3-p$ we get $3-p \mid 7(\mathbf{1 p})$

It follows that $p \in\{2 ; 4 ; 10\}(\mathbf{1 p})$. Therefore $x \in\{12 ; 48 ; 300\}(\mathbf{0.5 p})$