Problem Solving: the Locker Problem

An impending headache to the administrator in planning the locker operation in SST. He seeks your advise on how to resolve this issue:

Here is the problem:

In SST, there is a row of 100 closed lockers numbered 1 to 100. A student goes through the row and opens every locker. A second student goes through the row and for every second locker if it is closed, she opens it and if it is opened, she closes it. A third student does the same thing for every third, a fourth for every fourth locker and so on, all the way to the 100th locker. 

source:  seas.gwu.edu

The goal of the problem is to determine which lockers will be open at the end of the process.

TASK 1

Working in pairs, explain your thinking to the following problems clearly. Be sure to use appropriate mathematical language and methods. Post your answers in the comment and indicate both of your names.

(a) Which lockers remain open after the 100th student has passed?

(b) If there were 500 students and lockers, which lockers remain opened after the 500th student has passed?

TASK 2 (similar problem)

3 droplets of water fell at the following rate, droplet A at every 5 minutes interval, droplets B at every 12 minutes interval and droplets C at every half an hour interval. 

source: unreasonablydangerousonionrings.blogspot.co

(c) When do you think all the droplets, that is A, B and C will fall at the same time on the ground? 
(d) Identify at least 2 methods to solve this problem. 
(e) Is there a particular topic in maths that analyses such problems?