Question 134918: I have been given this problem for extra credit could someone please help with the solution.
There are 100 open lockers. There are 100 students.
(Note definition- Status change: If a locker is closed it becomes open or if the locker is open it becomes closed).
The first student changes the status of all the lockers (Student closes 100 lockers).
The second student changes the status of every two lockers (lockers 2,4,6,8...100 are opened).
The third studen changes the status to every third locker (the status of lockers 3,6,9,12....99 are changed).
This process is repeated until all 100 students have had their turn at changing the status of their locker(s).
Question: What lockers are left closed?
Found 2 solutions by solver91311, vleith:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! A locker will change state whenever the ordinal number of the student passing evenly divides the number of the locker. So, if there are an even number of even divisors, counting 1 and the number of the locker itself, (i.e., student 33 in front of locker 33), then the state of the locker will be the same as its state at the beginning. On the other hand, if there are an odd number of divisors, then the state will change an odd number of times and the state of the locker will be opposite to the state at the beginning.
So if a number has a an even divisor, then there is a quotient that is also an even divisor. For example, 6 divided by 2 is 3, and 6 divided by 3 is 2. Therefore, even divisors always come in pairs EXCEPT for perfect squares. Perfect squares have one divisor, namely the square root of the perfect square, that gives a quotient that is the same number as the divisor, hence that divisor can only be counted once. So perfect squares have an odd number of divisors, and every other positive integer has an even number of divisors.
That means that if every locker is open at the start, the lockers with numbers that are perfect squares, (1, 4, 9, 16...) will be closed, and the rest will be open.
Answer by vleith(2983)  (Show Source):
You can put this solution on YOUR website! For problems like these, it is often helpful to scale small and look for patterns
I suggest you work the problem with only 10 lockers and 10 students, and then with 20 and 20.
Pay attention to the pattern you see. If you still can't figure it out, send me an email and I'll give you a hint.
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