Christmas Contest - Team Round - Problem 23

There are 24 students numbered 1,2, ... ,24 such that student i is only friends with students 2i and 2i+1. Alex wishes to distribute toys to the students but he only has five different choices for the color of the toy.

If N is the number of ways he can distribute the toys such that every pair of friends must have a different color toy, find the number of factors of N.