AMC 12 2015a Problem 24

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that $$(\text{cos}(a\pi)+i\text{sin}(b\pi))^4$$ is a real number?

\textbf{(A)}\ \frac{3}{50} \textbf{(B)}\ \frac{4}{25} \textbf{(C)}\ \frac{41}{200} \textbf{(D)}\ \frac{6}{25} \textbf{(E)}\ \frac{13}{50}