1000 Students - Odd Lockers Open

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SUMMARY

The problem involves 1000 students interacting with 1000 lockers, where each student toggles the state of lockers based on their student number. The first student opens all lockers, while subsequent students toggle lockers that are multiples of their respective numbers. After all 1000 students have completed their actions, only the lockers corresponding to perfect square numbers remain open. This conclusion is derived from the fact that a locker is toggled for each divisor it has, resulting in an open state only for lockers with an odd number of divisors, which occurs for perfect squares.

PREREQUISITES
  • Understanding of basic number theory, specifically divisors
  • Familiarity with the concept of perfect squares
  • Knowledge of mathematical problem-solving techniques
  • Ability to analyze patterns in sequences and series
NEXT STEPS
  • Study the properties of divisors and their relationship to perfect squares
  • Explore mathematical proofs related to the toggling problem
  • Learn about combinatorial problems and their applications
  • Investigate similar problems involving toggling or flipping scenarios
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Students studying mathematics, educators teaching number theory, and anyone interested in combinatorial logic problems.

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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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