The Unit Circle and Circular Functions · Superchallenge problem

For a real number $t$, let $P(t)$ be its terminal point on the unit circle. For how many values of $t$ with $0\le t<2\pi$ are $P(t)$, $P(3t)$ and $P(5t)$ three distinct points forming a triangle with an angle of exactly $120^\circ$?