first student opens every locker. The second student closes all the lockers that are multiples of 2. The thirdstudent changes (closes an open locker or opens a closed one) all multiples of 3. The fourth student changesall multiples of 4. And so on. After all students have finished with the lockers, how many lockers are closed andwhich ones?

It will be the perfect squares. There are 10 perfect squares less than or equal to 100.

The parity (openness or closedness) of a locker m changes 1 time for each factor of m . Thus only those who have an odd number of factors remain closed. But to have an odd number of factors you need to have a factor whose cofactor is itself. ie; p is a factor because p*p=m. But this is iff m is a perfect square.