How Many Lightbulbs Will Remain Turned On?

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SUMMARY

The discussion revolves around a problem involving 1000 light bulbs and 1000 students, where each student toggles the state of specific light bulbs based on their student number. The first student turns all bulbs on, the second student turns off every even-numbered bulb, and subsequent students toggle bulbs at intervals corresponding to their student number. The conclusion reached is that 50 light bulbs remain turned on after all students have completed their actions.

PREREQUISITES
  • Understanding of basic mathematical concepts, particularly factors and multiples.
  • Familiarity with toggle operations and their effects on binary states (on/off).
  • Knowledge of sequences and series, particularly in relation to counting problems.
  • Basic problem-solving skills in combinatorial mathematics.
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  • Study the concept of factors and how they relate to toggling operations.
  • Explore combinatorial mathematics to understand similar problems involving switches and states.
  • Learn about mathematical induction as a method to prove results in similar scenarios.
  • Investigate other classic problems in number theory that involve toggling or flipping states.
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Mathematicians, educators, students studying combinatorial problems, and anyone interested in logical reasoning and problem-solving techniques.

nabliat5
there is a 1000 light bulbs in a room
near each bulb there is a switch which turns the bulb near it on in a push of a button
and turns it off in a second push of a button and turns it on again in the third push of a button.

1000 students come into the room
the first student pushing every button once (thus turning every lighbulb on)
the second student pushes every second button (turned off every even place lightbulb)
the third student pushes every third button
etc..

how many lightbulbs will remain turned on
if all 1000 students will act the same way as before(the 4th will push every 4th button etc..)
 
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uhm...50.
 

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