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Suppose that you have two gears as shown below, one fixed and the other free to rotate around the fixed one. Each gear has some number of teeth on it; let's call the number of teeth on the fixed gear n, and the number of teeth on the rotating gear m. Put a red dot on one of the teeth on the rotating gear, and a second dot by whatever slot on the fixed gear your dotted tooth happens to be resting in at the moment.

Now let the free gear rotate around the fixed one however many times is necessary for the dots to come back into line again. Clearly, if m = n then the free gear will only have to go around once, but otherwise it may have to go around many times before the dots come back into line.
Today's challenge: What single condition on m and n is both sufficient and necessary to ensure that the dotted tooth on the rotating gear visits every other slot on the fixed gear before coming back to rest in the dotted slot?
Bonus challenge: Given any m and n, calculate the number of times the free gear must rotate around the fixed one before returning to the starting position.


Solution to yesterday's challenge

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