MHB 1000 Students - Odd Lockers Open

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After all 1000 students have entered the school, the lockers that remain open are those numbered with perfect squares between 1 and 1000. This occurs because each locker is toggled (opened or closed) based on the number of its divisors, which is odd only for perfect squares. The first student opens all lockers, while subsequent students toggle lockers based on their number's divisibility. Ultimately, only lockers numbered 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961 remain open. The final answer confirms that only the square numbers are left open.
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Following recess, the 1000 students of a school lined up and entered the school as follows: The first student opened all of the 1000 lockers in the school. The second student closed all lockers with even numbers. The third student “changed” all lockers that were numbered with multiples of 3 by closing those that were open and opening those that were closed. The fourth student changed each locker whose number was a multiple of 4 and so on. After all 100 students had entered the building in this fashion, which lockers were left open?
 
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Re: word problem for my math assignment

I think you mean after 1000 students. If so, the answer is all the square numbers between 1 & 1000.
 

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