A high school has 1000 students and 1000 lockers, one locker for each student. On the first day of school, the principal plays the following game: She asks the first student to go and open all the lockers. She then asks the second student to go and close all the even-numbered lockers. The third student is asked to check every third locker. If it is open, the student closes it; if it is closed, the student opens it. The fourth student is asked to check every fourth locker. If it is open, the student closes it; if it is closed, the student opens it. The remaining students continue this game. In general, the nth student checks every nth locker. If the locker is open, the student closes it; if it is closed, the student opens it. After all the students have taken their turn, some of the lockers are open and some are closed. Write a program that prompts the user to enter the number of lockers in a school. After the game is over, the program outputs the number of lockers that are opened. Test run your program for the following inputs: 1000, 5 0 0 0, 10 0 00. Do you see any pattern developing?
(Hint: Consider locker number 100. This locker is visited by student numbers 1, 2, 4, 5, 10, 20, 25, 50, and 100. These are the positive divisors of 100. Similarly, locker number 30 is visited by student numbers 1, 2, 3, 5, 6, 10, 15, and 30. Notice that if the number of positive divisors of a locker number is odd, then at the end of the game the locker is opened. If the number of positive divisors of a locker number is even, then at the end of the game the locker is. closed.)
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