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1000 Locker Problem

At Thousand Locker High, there are exactly 1000 students and 1000 lockers. It's the first day of school, and all 1000 lockers are initially open. The first student walks down the hall (beginning at the first locker) and reverses the position (open/close) of each locker. Now all the lockers are closed. Next, the second student walks down the hall and reverses the position of every second locker. Now all the even numbered lockers are open. Next, the third student walks down the hall and reverses the position of every third locker. Next, the fourth student walks down the hall and reverses the position of every fourth locker and so forth until all 1000 students have walked down the hall with the 1000th student reversing only the position of the 1000th locker. How many lockers are now closed?

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