Question

At Gratis High School, there are 1, 000 students, numbered 1, 2, 3, . . . , 1, 000. The very long main hallway at Gratis as 1, 000 lockers along the wall, also numbered 1, 2, 3, . . . , 1000. On the first day of school all the lockers are closed.
The students arrive one at a time and walk the length of the main hallway, opening and closing lockers according to the following rules:
1. Student 1 opens every locker.
2. Student 2 closes every second locker.
3. Student 3 changes the state of every third locker. By change the state, we mean that
a student closes the locker if it is open, and op

Answer 1

At Gratis High School, there are 1, 000 students, numbered 1, 2, 3, . . . , 1, 000. The very long main hallway at Gratis as 1, 000 lockers along the wall, also numbered 1, 2, 3, . . . , 1000. On the first day of school all the lockers are closed.
The students arrive one at a time and walk the length of the main hallway, opening and closing lockers according to the following rules:
1. Student 1 opens every locker.
2. Student 2 closes every second locker.
3. Student 3 changes the state of every third locker. By change the state, we mean that
a student closes the locker if it is open, and opens the locker if it is closed.
k. Student k changes the state of every kth locker.
After all 1,000 students have arrived for school, which lockers are open and which lockers are closed?

Answer 2

@ganeshie8 i know that the perfect squares are the lockers left open because we discussed this in class, but how would I be able to answer: Which lockers are touched by both Students 6 and 8? What is the first locker touched by both?