Open, close, open, close, open......Number Theory Level 2
A royal palace has \(1000\) rooms numbered \(1\) to \(1000\). Each room is maintained by a cleaner bearing the room number. Each morning, cleaner number \(1\) will open all the room doors. Cleaner number \(2\) will close all the doors of room with numbers divisible by \(2\). Cleaner number \(3\) will then approach rooms with numbers divisible by \(3\). If the room door is opened, he will then close it, if the door is closed, he will then open it. Cleaner number \(4\) will take care of all the rooms with numbers divisible by \(4\). He will open closed doors and close opened doors.
This process goes on until every cleaner(\(1000\) cleaners) has done his job. At the end, how many room doors remained open?