MathDB

N

cards are arranged in a circle, with exactly one card face up and the rest face-down. In a turn, choose a proper divisor

k

N

. You may begin at any card on the circle and flip every

k

-th card, counting clockwise, if and only if every

k

-th card begins the turn in the same orientation (either all face-up or all face-down).
For example, with

15

cards, you may start at any position and flip the

3

rd,

6

th,

9

th,

12

th, and

15

th cards around the circle if they all begin the turn face up (or all face-down).
For what values of

N

can all of the cards be flipped face-up in a finite number of turns?

2