Finite automata that accept binary strings divisible by n
There is a known formula for minimum number of states for such a finite automaton. This depends on $n$ as well as the radix $R$ of the underlying positional representation. 

If $n$ is coprime to $R$, then the minimal number of states is $n$. However, when $n$ shares a factor with the radix then the situation is rather complicated. See Mathematica Journal Vol 3 Issue 11. "Divisibility and State Complexity" by Klaus Sutner.