Circular Room

Number Theory Level pending

9 people are assembled in a circular room with 17 doors numbered 0 through 16 inclusive which all start off closed. Then, the first person goes through and switches every door. The next person goes through and closes a door if open and opens it if closed for the \(3^\text{rd}\) powers mod 17. The next person does the fifth, and so on up until \(17^\text{th}\) powers. Which doors will be open and closed?

Express your answer as seventeen 1s or 0s where a 1 represents an open door and a 0 a closed one.

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