2015 AMC 12A 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} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.