AMC 8 2008 Problem 24

Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? $\textbf{(A)}\ \frac{1}{10} \textbf{(B)}\ \frac{1}{6} \textbf{(C)}\ \frac{11}{60} \textbf{(D)}\ \frac{1}{5} \textbf{(E)}\ \frac{7}{30}$