There are 1000 doors in a palace which also contains 1000 servants. Every servant has a distinct tag numbered from 1 to 1000. Everyone has a perfect work set; i.e. to operate the doors taking their tag no.'s multiple. In simple words, man1 will open all doors.. Man2 will close even numbered doors.. Man3 will open closed doors and will close opened doors that are numbered as multiple of 3. And so on till 1000th servant operates 1000th door. Now, how many doors are closed at last.
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1That's a famous riddle. Just think about how many times door number $n$ gets changed (hint: look at the prime factorization). – Mathematician 42 May 09 '17 at 06:23
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31 doors are open at last and remaining are closed. – Sameed_Naama May 09 '17 at 07:24