2007 AMC 10B Problem 25

How many pairs of positive integers (a,b) are there such that a and b have no common factors greater than 1 and \frac{a}{b} + \frac{14b}{9a} is an integer?

\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } \text{infinitely many}

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