Vocaloid:

100 Doors Riddle (for fun)

Vocaloid:

There are 100 closed doors, each labelled from 1 to 100. There are also 100 people. Person #1 goes through every door and opens it. Person #2 goes through every second door and closes it. Person #3 changes the status of every third door (that is, he will open the closed doors and close the open doors. Person #4 changes the status of every fourth door. This continues until Person #100 has finished. In the end, which doors are open?

Vocaloid:

For clarity: Person #1 changes the status of doors 1,2,3,4, etc. Person #2 changes the status of doors 2,4,6,8, etc. Person #3 changes the status of doors 3,6,9, etc.

Vocaloid:

Hint: the answer has something to do with factorization

Tranquility:

Hmm. So at the end, doors with even factors will remain closed while doors with odd factors will remain open. And most numbers have even factors. The only numbers that have odd factors are perfect squares because one of the factors repeat. Therefore, the doors that remain open are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100

Vocaloid:

^ perfect

Tranquility:

It took a while to think of it, thanks for the riddle!