Language of binary strings divisible by 7
The langauge is regular, so answers 1&ndash;4 are all correct.

The (deterministic) automaton has seven states, $0, \dots, 6$ corresponding to the remainder when the number is divided by&nbsp;7. The initial state is&nbsp;$0$, which is also the only accepting state.  We adopt the convention that the empty string represents zero (as does the string "$0$", obviously).

To describe the transition function, suppose you're reading in some binary string and the bits you've read so far correspond to the natural number&nbsp;$n$ and the state you're in is $r=n\%7$ (the remainder when $n$&nbsp;is divided by&nbsp;$7$). Let the next bit be&nbsp;$b\in\{0,1\}$.  When you read that, you'll have seen the number $2n+b$, so you need to move to state

$$r' = (2n+b)\%7 = (2r+b)\%7\,.$$

And, of course, there's nothing special about&nbsp;$7$.