1. At Mathschool, the staff have a very odd tradition:
After each of the 100 pupils has been given their locker number, they all, in turn, walk down the whole row of 100 lockers, and any lockers with numbers that are a multiple of their number they either open it if it’s already closed, or close it if it’s already open.
If all the lockers are closed at the beginning, how many are open after the 100th person has finished?
Hint:
consider one particular locker – what condition must be met for it to be open at the end?

2. How many times will these be opened or closed?4 5 6 7 8 9
1, 2, 3, 6
1, 2, 4, 8
1, 2, 4
1, 5
1, 7
1, 3, 9