The Locker Problem>
Algebra
Level
2
One hundred students are assigned lockers 1 through 100.
- The student assigned to locker number 1 opens all 100 lockers.
- The student assigned to locker number 2 then closes all lockers whose numbers are multiples of 2.
- The student assigned to locker number 3 changes the status of all lockers whose numbers are multiples of 3 (e.g. locker number 3, which is open, gets closed; locker number 6, which is closed, gets opened).
- The student assigned to locker number 4 changes the status of all lockers whose numbers are multiples of 4, and so on for all 100 lockers.
Which lockers would stay open?
No lockers
Odd-numbered lockers
Even-numbered lockers
Prime-numbered lockers
Square-numbered lockers
All lockers
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1 solution
Define f(n) is the positive dividers of n.Locker number n will be opened or closed for f(n) times.If locker number n is open at last,then f(n) is an odd number,this means n is a perfect square.