20 boxes
There are 20 boxes numbered 1-20 and 20 students with IDs 1-20. All the boxes are initially closed. The student whose ID is 1 then opens all the boxes. The second student then closes box number 2,4,6 ---- 20. The third student then opens Box 3, closes box 6 and so on. Likewise, all students perform an operation on only those boxes, the numbers of which are multiples of their IDs. If a box is opened, the student will close it and if it is closed, he/she will open it. After all 20 students perform their respective tasks, how many boxes will remain open?
The answer is 4.
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1 solution
If a box's number is k , then it will change, when a person's ID is a divisor of k . A box will be open at the end if and only if it changes an odd number of times, ie. the box number has odd number of divisors. If a number has odd number of divisors, then it is a perfect square. The perfect squares at most 20 are 1,4,9,16.
Therefore the answer is 4 .