$x,y\times xy,x=xy,xy$

$\Leftrightarrow \frac{\overline{xy}}{10}\times \left( \overline{xy}+\frac{x}{10} \right)=\overline{xy}+\frac{\overline{xy}}{100}$

$\Leftrightarrow \frac{\overline{xy}}{10}\times \left( \overline{xy}+\frac{x}{10} \right)=\frac{\overline{xy}}{10}\times \frac{101}{10}$

$\Leftrightarrow \overline{xy}+\frac{x}{10}=\frac{101}{10}$

$\Leftrightarrow 10x+y+\frac{x}{10}=\frac{101}{10}$

$\Leftrightarrow 101x+10y=101$

$\Rightarrow x=1,y=0$