NUMBER PROBLEM Suppose that a school has 1,000 lockers and that they are all closed. A person passes through, opening every other locker, beginning with locker 2. Then another person passes through, changing every third locker (closing it if it is open, opening it if it is closed), starting with locker 3. Yet another person passes through, changing every fourth locker, beginning with locker 4. This process continues until 1,000 people pass through.
(a) At the end of this process, which locker numbers are closed?
(b) Write an explanation for your answer to part (a). (Hint: It may help to attempt exercise 34 first.)