Question 10
There is a unique angle $\theta$ between $0^\circ$ and $90^\circ$ such that for nonnegative integers $n,$ the value of $\tan(2^n\theta)$ is positive when $n$ is a multiple of $3,$ and negative otherwise. The degree measure of $\theta$ is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$