100 doors start closed, numbered 1 to 100

100 doors start closed, numbered 1 to 100. Over 100 rounds: in round k, every door whose number is evenly divisible by k gets toggled (e.g. round 3 toggles doors 3, 6, 9, 12...). Ananya thinks that after 100 rounds, a random-looking scattering of doors will be open, with no clean pattern. Her friend says the open doors will follow a clean mathematical pattern. Who is right?