Solve High School Lockers Calculus Problem

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    Calculus
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SUMMARY

The high school lockers problem involves 1000 lockers and students toggling the state of lockers based on divisibility rules. Each student opens or closes lockers that correspond to their locker number's divisors. The solution reveals that only the lockers with perfect square numbers remain open at the end, specifically lockers 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. This results in a total of 31 open lockers.

PREREQUISITES
  • Understanding of basic number theory concepts, particularly divisibility.
  • Familiarity with mathematical patterns and sequences.
  • Ability to analyze and derive conclusions from numerical data.
  • Basic problem-solving skills in calculus or algebra.
NEXT STEPS
  • Explore the concept of perfect squares and their properties.
  • Learn about mathematical induction as a proof technique for similar problems.
  • Investigate combinatorial problems involving toggling states or binary operations.
  • Practice solving similar problems involving divisors and their effects on sequences.
USEFUL FOR

Students in mathematics, educators teaching problem-solving techniques, and anyone interested in combinatorial logic and number theory applications.

whisperblade
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Hi everyone, i recently got this problem from my college professor and I am either incredibly rusty or just don't know how to do this. I've tried to set up some sort of equation using f(x) but i just can't make anything fit or account for everything. any help would be appreciated.

the question follows:
A high school has 1000 students and each has a numbered locker where they keep various smelly items. Fortunately all the locker doors are shut. One by one, each student walks past the lockers, and either opens or shuts (depending on its previous position) the door of any locker that divides their own locker number. How many lockers are open at the end?
 
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My daughter got this same problem a few weeks back. Suggest you just plug-n-chug to map out the 1st 15-20 students until you see the pattern. At least, that's how we did it.
 

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