For what values of $a$ and $b$ (where $a$ and $b$ are positive integers) do the two equations:

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{x^2-ax+b}& =0\\ \multicolumn{1}{c}{x^2-bx+a}& =0\end{array}}$

both have positive integer solutions? You may be able to find some values of $a$ and $b$ by trial and error. Can you prove that these are the only possible values?