Bernard's Cars Problem: Exit even-numbered cars, then every multiple of 3, etc

jtayag0622

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Sep 10, 2018
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Please help! I don't know how to solve this problem:

Bernard, who is a genius and very wealthy, owns one thousand cars, each of which is numbered with a different natural number from 1 to 1000. He invites 1000 people to a party, and he makes each person sit in a different car so that every car is occupied. He then tells all the people who are in even numbered cars to get out of their cars. then he tells everyone with a car numbered any multiple of 3 to get out (or get in if they are already out of it). Then he tells everyone with a car numbered any multiple of 4 to get out (or get in). He continues giving these instructions until he tells the last instruction: anyone with a car numbered a multiple of 1000 should get out of the car (or get into it). After all these instructions are given which cars will be unoccupied? How many occupied cars? Why those cars?
 
Did you see a pattern, when you performed some initial steps on paper?

I wrote out the first 24 numbers and went through some steps. I think I see a pattern.
 
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