2007 AMC 12B Problem 24

How many pairs of positive integers (a,b) are there such that \gcd{(a,b)}=1 and \frac{a}{b}+\frac{14b}{9a} is an integer?

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

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