Lexical Number Theory

Computer Science Level 4

Assume that the strings accepted by the automata below are interpreted as binary numbers.

What is the largest integer $$n$$ such that the set of integers accepted by the automata are always divisible by $$n$$?

Extra Credit: Prove your answer.

Extra Extra Credit: Is it always possible to form such an automata, for any $$n$$? If not, why not? If yes, how?

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