The Locker Problem Extended

Number Theory Level 4

Every day, 100 students enter a school that has 100 lockers. All the lockers are closed when they arrive.

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 \(n^\text{th}\) locker.

One day, however a few of the 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 1 is open and all the other 99 lockers are closed.

How many students were absent that day ?


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