There are 10,000 lockers.
One person opens all the lockers.
The second person closes every 2 doors.
The third person counts every three locker and does the opposite to the locker. (Open/close it.)
The fourth person counts every fourth locker and does the opposite to it.
The fifth person counts every fifth locker and does the opposite and so on.
By the time the 10,000th person gets to the 10,000th lockers, would it be closed or opened?
Sorry for putting this up again.
Hint:
The factors of a number, say 12, are
1,2,3,4,6,12, or
rearranged differently,
1,12, 2,6, 3,4
for a total of 6 (even).
The factors of a prime number, say, 23, are
1,23
for a total of 2 (even).
The factors of another composite, say, 36, are
1,2,3,4,6,9,12,18,36, or rearranged,
1,36, 2,18, 3,12, 4,9, 6
for a total of 9 (odd).
Can you figure out which lockers are open, and why?