Take the 2-minute tour ×  Suppose you need to slow down a
turning motor so that a gear turns at an angular velocity
$\frac{a}{b}$ of that of the motor shaft, where $a$ and $b$ are
natural numbers. For example, this set of four gears achieves a
slowdown of $\frac{1}{60}$: the $10$ teeth of the motor green
gear are slowed to $\frac{1}{6}$ by the $60$ teeth of the blue
gear, and the $10$ teeth of the red gear are slowed another
$\frac{1}{10}$ by the $100$ teeth of the gray gear:           
(Image from AMS Feature Column.)  There are two natural costs to
a gear train: the teeth cost, the total number of teeth
(assuming the manufacturing costs are dominated by the teeth),
and the metal cost, the total area of the gears (assuming the
cost of the metal dominates). Since the area is determined by
the radii, which are proportional to the number of teeth, let us
say the metal cost is the sum of the squares of the number of
teeth. So the gear train above has teeth cost $10 + 60 + 10 +
100 = 180$ and metal cost $10^2 + 6