Number Theory with Finite State Machines

Assume that the strings accept 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 be divisible by \(n\)?

Clarification: Empty string is considered as equivalent to 0 in this question.

Extra Credit: Both this FSM and the one in Agnishom Chattopadhyay's question (linked below) has three states. But the answers are different. Account for this phenomenon.

Inspiration: Lexical Number Theory


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