Question about mapping onto the unit circle.

Question about mapping onto the unit circle.

Let $K$ be a positive integer and let $x$ be an angle in $(0,2ð)$.
Consider the map $g: {0,1,..K-1,K}$ onto the unit circle, given by
$g(n)=n*x (mod 2ð)$. Prove the image of $g$ divides the circle into arcs
of one, two, or three different lengths.
I can see that the arc lengths are equal whenever $x$ is of the form
$\dfrac{ð}{b}$, where $b$ is an integer, but I'm confused on how to go
about the rest. Any hints would be greatly appreciated.