Minimal DFA that accepts binary strings whose decimal equivalent is divisible by 32
Binary numbers that are divisible by 32 are ending in zero's. You can just keep track of how many consecutive zero's we have seen. If you want to answer the same question for the decimal alphabet, yes, then you need to worry about remainders modulo 32. The phrase "decimal equivalent" is a bit confusing here: divisibility of a number does not change when choosing another representation. 

After building the automaton you can minimize it using standard constructions, to see if you cannot do with less states. Perhaps the same approach works for $2^n$, but then you not have a single "concrete" automaton. Alternatively you might use the theory of Myhill-Nerode equivalence classes of the language, to show the number of states is at least a certain number.