I was dealing with a childish problem, which eventually led to this:
Find integer $n$ and $m$ to fit the equation $6n=7m+1$
I immediately found first numbers by simple enumeration ($n=6$, $m=5$). But later I spent half of the day to figure out whether it's possible to have some general solution to the problem or at least any solution without manual enumeration.. I found nothing. So is it possible? Thanks!