2009 AMC 10A Problem 19

Circle A has radius 100. Circle B has an integer radius r<100 and remains internally tangent to circle A as it rolls once around the circumference of circle A. The two circles have the same points of tangency at the beginning and end of circle B's trip. How many possible values can r have?

\mathrm{(A)}\ 4 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 50 \qquad \mathrm{(E)}\ 90

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