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 Math Help > Math Puzzles > 2004 State Logic Quiz > Question 05 > Question 05

# Academic Decathlon 2004 State Championship Logic Quiz

## Question 05

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?

65

In order to get the "13" wheel back to its original position, some multiple of 13 teeth must pass a fixed point.
In order to get the "6" wheel back to its original position, some multiple of 6 teeth must pass a fixed point.
In order to get the "5" wheel back to its original position, some multiple of 5 teeth must pass a fixed point.
Likewise, in order to get the "2" wheel back to its original position (neglecting the practical difficulties of getting a two-toothed wheel to work properly at all!), some multiple of 2 teeth must pass a fixed point.  Combining the requirements wheels 13, 6, 5, and 2, we find their Least Common Multiple (LCM).

LCM(13,6,5,2) = 390

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.

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