Case of Open Locker Number 4...Number Theory Level 5
Student \(1\) opens every locker.
Student \(2\) closes every second locker.
Student \(3\) changes the state of every third locker i.e. he opens it if it is closed and closes it if its open.
Student \(4\) changes the state of every fourth locker and so on... so that student \(n\) changes the state of every \(nth\) locker.
One day, on account of a blizzard, several students are absent. Regardless, those present complete the procedure and simply skip the students who are absent. For e.g. if student \(3\) is absent, then nobody changes the state of every third locker.
At the end of the process, it is found that only locker number \(4\) is open and all the other \(99\) lockers are closed...
How many students were absent that day ?
More interestingly, given a locker number n, find a general rule that keeps only locker n open and all other lockers closed.