There are 100 seniors in Fishtopia High School. Each senior has their own locker numbered #1-100 down a long corridor. As a graduation tradition, all the seniors line up and walk one at a time down the hallway. The first senior changes all the locker positions (so, the first person in line opens all the locker doors). The second senior then changes the position of every other locker (so, since all the lockers are now open, she closes door #2, closes door #4 etc. while not touching door #1 or door #3, etc.). The third senior then changes the position of every third locker (so, he closes door #3, opens door #6, etc.). This continues until all seniors have had an opportunity to walk down the corridor, only changing the position of the locker doors that correspond with multiples of their position in line (so, the last senior only changes the position of locker #100, while not touching any of the other lockers). [Note: Changing the position of a locker means opening it if it is closed or closing it if it is open.]

a. [3 pts] How many students touched locker #25? Is this locker open or closed at the very end after all 100 seniors have walked down the corridor?

b. [2 pts] At the very end, only ten of the lockers will be open. Which lockers will be open at the very end? What do these lockers have in common?