Challenge Level

For what values of $a$ and $b$ (where $a$ and $b$ are positive
integers) do the two equations: $$x^2-a x+b=0$$ $$x^2-b x+a=0$$
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?