Children and light switches

[I got this problem from Mariela Pavlova.]

A room has 100 light switches, numbered by the positive integers 1 through 100. There are also 100 children, numbered by the positive integers 1 through 100. Initially, the switches are all off. Each child k enters the room and changes the position of every light switch n such that n is a multiple of k. That is, child 1 changes all the switches, child 2 changes switches 2, 4, 6, 8, , child 3 changes switches 3, 6, 9, 12, , etc., and child 100 changes only light switch 100. When all the children have gone through the room, how many of the light switches are on?

©2019 K.R.M. Leino - Split Template by One Page Love