Given a circle and n points in it such that these points are dividing the circle into equal parts. There is also a positive number k which can be less/more than n. It is allowed to move (only clockwise or only counterclockwise) across the perimeter of the circle by a step equal to k started from some point n(i). How it can be proven, that always there is a number m such that after m*k moves one can return to the starting point n(i)?
My approach is the following: for any n and k, always exists lcm(n, k). It means that we can write the following expression: cn = mk or lcm(n, k) = mk . This is, in turn, means that mk contains at least one complete n so at some point it has been reaching the starting point. But I'm not sure about this approach.