The number of times the ~i^{th}~ ring turns depends only on its radius and the radius of the first ring. The sought ratio is ~R_i / R_1~. Reducing the fractions can be done by dividing the numerator and denominator by their greatest common divisor, given by the recursive formula:

$$\displaystyle \gcd(a,b) = \begin{cases}a & \text{when }b=0 \\ \gcd(b, a \bmod b) & \text{when }b \ne 0\end{cases}$$