1976 AHSME Problem 8

A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?

\textbf{(A) }\frac{13}{81}\qquad \textbf{(B) }\frac{15}{81}\qquad \textbf{(C) }\frac{13}{64}\qquad \textbf{(D) }\frac{\pi}{16}\qquad \textbf{(E) }\text{the square of a rational number}

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