A school has 1000 students and 1000 lockers Student N is ass

A school has 1000 students and 1000 lockers Student N is assigned to locker N. One day student 1 comes to school and opens all 1000 lockers. Student 2 then comes into school goes to her locker (locker #2) and closes it Student 2 then decides to also close lockers 4, 6, 8, 10,... etc Student 3 corned into school goes to his locker (#3) and also lockers 6, 9, 12, 15,...etc and if a locker is open he closes it. and if a locker is closed he opens it. Student 4 goes to lockers 4, 5, 12, 16,... etc and opens the locker if it\'s closed, and closes the locker is it\'s open. Students goes to locker n and all subsequent multiples of n and undoes whatever state the locker is in. If this pattern continues through all 1000 students, which lockers remain open after the 1000* student has gone through?

Solution

the numbers which are squares are remain open afetr 1000th student gone through, i.e 1,4,9,16,25,36,........

reason is

any number except squares are even no of divisors i.e

divisors of 2 is 1 & 2.

divisors of 3 is 1 & 3.

divisors of 6 is 1,2,3,6.

divisors of 15 is 1,3,5,15.etc

so one number open the locker and other number close it.since number of divisors even after last the locker must close.for an example suppose number is 15.1 open it,3close it,5open it & 15 close it.

for squares no of divisors is odd always i.e

divisors of 1 is 1.

divisors of 4 is 1,2,4.

divisors of 16 is 1,2,4,8,16.and so on.

so one open it,second close it & last one obviously open it again.for an example suppose the number is 16.1open it,2close it,4open it,8close it & 16 open it.