A number theory problem by Joshua Cortright
Elliot had 100 light bulb switches. He invited 100 friends over and assigned them a different number from 1 to 100. All of the switches were off. The person who was number 1 flipped every 1st switch, the person who was number 2 flipped every 2nd switch, and so on until the person who was number 100 flipped every 100th switch. How many switches were on at the end?
The answer is 10.
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1 solution
The n -th bulb has been flipped for d ( n ) times, where d ( n ) is the function counting number of divisors of n .
A bulb is on only if it has been flipped for odd number of times, and only prefect square has odd number of divisors.
There are 1 0 prefect square from 1 to 1 0 0 , so the answer is 1 0 0 .