The diagram, below, illustrates four gears, or toothed wheels. The
number shown inside each wheel tells the number of teeth around the outside of
the wheel. How many turns must the second-largest wheel make to bring
every wheel back to its original position?
In order to get the "13" wheel back to its original position, some
multiple of 13 teeth must pass a fixed point.
So we need to turn wheel 6 enough times to make 390 teeth pass a fixed point. Each turn of wheel 6 causes 6 teeth to pass a fixed point, so we need 390/6 = 65 turns of this wheel to bring all the wheels back to their original position.
See the NEXT puzzle in the Academic Decathlon 2004 Logic Quiz
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