Let's say you have a positive integer $x$ expressed in a positive integer base $b$. You desire to test if $c$, another positive integer, divides $x$. Well, if $c$ divides $b$, then then, if the number, $x$ expressed in base $c$ has a last digit which is divisible by c (mod c = 0), then $c$ divides $x$.

How is this useful? Dunno. It can be used for super large numbers, and I think it can also be used recursively, though. Maybe it's trivial to you, but just thought it would be fun to share. People rarely ever count in other bases, and I feel there's untapped potential for mathematics there.