59th Ukrainian National Mathematical Olympiad

Let us rewrite this number as follows: $$\overline{30x070y03}=300070003+10^{6}x+10^{2}y=37\cdot (8110000+27027x+3y)+(3+x-11y).$$ Thus, $3+x-11y$ must be divisible by $37$ for the smallest $x$. Since $x$, $y$ are digits, then this expression can only take values $0$, $-37$, $-74$. For each, we have the following answers. $$3+x-11y=0\Rightarrow y=1\text{ and }x=8.$$ $$3+x-11y=-37\Rightarrow y=4\text{ and }x=4.$$ $$3+x-11y=-74\Rightarrow y=7\text{ and }x=0.$$ Thus, the smallest number of these is $300070703$.