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ankur17
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Bunuel
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A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
Note: The first locker that the n-th student changes is the n-th locker.

1. 500
2. 32
C. 31
D. 30
E. 250
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If you think of, say, locker #6, it gets flipped open or closed by the 1st, 2nd, 3rd and 6th student, i.e. it will be flipped by student n any time n is a divisor of 6. So locker #6 will stay closed, because 6 has an even number of divisors. On the other hand, locker #25 say gets flipped by the 1st, 5th and 25th student, so it will end up open because 25 has an odd number of divisors.

The only numbers with an odd number of divisors are perfect squares. So the question is just asking: "How many perfect squares are there between 1 and 1000 inclusive?" Now 32^2 might be familiar, since it's a power of 2 -- it is equal to 1024, so 32 is too big, and the answer is 31.

I'd add that in a real GMAT question, when answers are simple integers, the answer choices would always be listed in increasing or decreasing order, and real GMAT questions wouldn't waste time discussing how "strange" the principal of a high school might be, so this is clearly not an official question. I suspect if something similar were to appear on the real test (which seems unlikely to me), the number of 'lockers' involved would be far less than 1000.
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A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
Note: The first locker that the n-th student changes is the n-th locker.

1. 500
2. 32
C. 31
D. 30
E. 250

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