jmc

Rewrite $\underline{3\Theta }+\Theta$ as $30+\Theta +\Theta =30+2\Theta$, so we have an algebraic expression we can manipulate. Multiply by $\Theta$ to obtain: $$\begin{array}{rl}252/\Theta & =30+2\Theta \Rightarrow \\ 252& =30\Theta +2{\Theta }^2\Rightarrow \\ 0& =2{\Theta }^2+30\Theta -252\Rightarrow \\ 0& ={\Theta }^2+15\Theta -126\Rightarrow \\ 0& =(\Theta +21)(\Theta -6).\end{array}$$Or using the quadratic formula: $$\begin{array}{rl}\Theta & =\frac{-15\pm \sqrt{225-4\cdot 1\cdot -126}}{2}\Rightarrow \\ & =\frac{-15\pm \sqrt{729}}{2}\Rightarrow \\ & =\frac{-15\pm 27}{2}\end{array}$$Either way, because $\Theta$ must be a positive digit, $\Theta =\boxed{6}$.