At a school with 1000 students and 1000 lockers, Student 1 has locker 1 and so on. Student 1 decides to open all 1000 lockers. Student 2 then decides to go to every second locker and close it. Student 3 then goes to every third locker and if the locker is opened they close it. If it is closed, student 3 opens it. Student 4 changes the condition of every fourth locker; if it is opened student 4 closes it and if it is closed, student 4 opens it. The process continues all the way through student 1000 who changes the condition of the 1000th locker. In a world of N lockers which lockers stay open?

Question

campbell_st · Answer

the prime lockers

jim_thompson5910 · Answer

Student 1 opens locker 2

Student 2 closes locker 2

jim_thompson5910 · Answer

2 is prime, so it's not the prime lockers

campbell_st · Answer

well the composites

anonymous · Answer

so the composites?

anonymous · Answer

oh haha thanks!

jim_thompson5910 · Answer

more specifically, which composites?

campbell_st · Answer

oop.... 1 is neither prime nor composites... so it can't be composites...

jim_thompson5910 · Answer

this is a question for pepper0331 to think about

anonymous · Answer

oh... i tried to think about it. i don't know how to go about solving it

jim_thompson5910 · Answer

List the factors of the first few positive integers

1:   1
2:   1, 2
3:   1, 3
4:   1, 2,  4
5:   1, 5
6:   1, 2, 3, 6
7:   1, 7
8:   1, 2, 4, 8
9:   1, 3, 9

Notice anything?

anonymous · Answer

wait i dnt know what does that have to do with the question?

jim_thompson5910 · Answer

The factors of each locker are the students who open/close them

jim_thompson5910 · Answer

Ex: look at locker 8

student 1 opens locker 8
student 2 closes locker 8
student 4 opens locker 8
student 8 closes locker 8

jim_thompson5910 · Answer

Does that make sense?

anonymous · Answer

ohhh ya

jim_thompson5910 · Answer

So which lockers stay open?

mathmate · Answer

perfect ...

anonymous · Answer

wait i dnt know. the ones that end with 4?

jim_thompson5910 · Answer

Notice how 1 has only 1 factor

4 has 3 factors (1, 2, 4)

9 has 3 factors (1, 3, 9)

If you keep going...

16 has 5 factors (1, 2, 4, 8, 16)

25 has 3 factors (1, 5, 25)

So the rule is this: If a locker number has an odd number of factors, then it will stay open at the end.

All perfect squares have an odd number of factors. So all the perfect square lockers will stay open

anonymous · Answer

OH... thank you!

jim_thompson5910 · Answer

you're welcome