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?

1. $r$ must be a divisor of $150$.
2. $150=2\cdot3\cdot5^2$ has $\tau(150)=2\cdot2\cdot3=12$ divisors; excluding $r=150$ itself leaves $\boxed{11}$.