Question 235483: A very long hallway has 100 lights with pull cords hanging from the ceiling. All the lights are off. A person comes along and pulls every cord, turning on all the lights. A second person comes along and pulls every second cord. This would turn off light numbers 2, 4, 6 etc. A third person comes along and pulls every third cord. Turning on some lights and turning off others. This continues until the one hundredth person comes along and pulls the one hundredth cord. At this point, which lights are on?
Answer by solver91311(24713)   (Show Source):
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A person will only touch a light cord if that person's number is an even divisor of the lamp number.
If  is and even divisor of  , meaning that  and  is an integer, then must also be an even divisor of , meaning  and is an integer. That means even divisors always come in pairs -- hence there is always an even number of even divisors for every integer. With one notable exception -- perfect squares. Perfect squares have an odd number of even divisors because one of the pairs of divisors is such that  .
Try it:
 is not a perfect square and has 2 divisors, namely  and .
 is not a perfect square and has lots of divisors, namely ,  ,  ,  ,  ,  ,  , and . But there are 8 of them -- an even number.
 IS a perfect square. The even divisors are , , , , . Five even divisors -- an odd number.
If you start with a light off, and you operate the switch an even number of times, the light will be off. If you operate the switch an odd number of times, the light will be on. Hence, the perfect square number lights will be on and the rest will be off.
John
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