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$
Solution
Circle $B$ returns to the same point of tangency exactly when the circumference $2\pi\cdot100$ of circle $A$ is an integer multiple of the circumference $2\pi r$ of circle $B$.
So $r$ is a divisor of $100$. Since $\tau(100)=9$ and $r=100$ is excluded, there are $8$ values.
Circle $A$ has radius $150$. Circle $B$ has integer radius $r<150$ and rolls once around circle $A$, returning to the same tangency point. How many possible values can $r$ have?
