Maths Puzzle: Back to Black

I have a board that uses the numbers 1 to 90. I fill the board with Othello piece, they are black on one side and white on the other. All the pieces start black side up. First I flip every second number, so 2 becomes white, and 4, 6, 8 and so on. Then I flip every third number, then every fourth number, until finally I flip every 90th number. Which numbers end up black side up and why?

Music by Ghostly Dust Machine http://freemusicarchive.org/music/Gho...

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