2008 AMC 10B Problem 15

How many right triangles have integer leg lengths a and b and a hypotenuse of length b+1, where b<100?

\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 7\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 10

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