Question

This is how the problem starts. All the cbana doors start out being closed. Student assigned to cabana 1 opens the doors on all 100 cabanas. The students assigned to cabana #2then closes doors on all cabanas whose numbers are multiples of 2. Th e student assisgned to cabana #3 then changes the state of all cabanas whose numbers are multiples of 3 ex: the door on cabana #3 which is open gets closed, and cabana #6 which is closed get opened. Come up wit some sort of system to simulate the opening and closing of the doors

Ah, a restatement of the locker problem.

While not stated in your original problem, I presume you have 100 students, not just 3.

At the end of the day, the open doors will be those numbers that have an odd number of factors. These will only be numers that are perfect squares. (1,4,9,...). All other doors will be closed.