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I have a machine which has four sequential cog wheels in constant mesh.

The largest cog has 168 teeth and the others have 49, 32 and 15 respectively.

How many revolutions must the largest cog make before each of the cogs is back in its starting position?

If we break each wheel into its prime factors, we get:

168= 2 x 2 x 2 x 3 x 7
49 = 7 x 7
32 = 2 x 2 x 2 x 2 x 2
15 = 3 x 5

So

revolutions x 168
---------------------------------
2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 7

revolutions x 2 x 2 x 2 x 3 x 7
---------------------------------
2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 7

revolutions
-------------
2 x 2 x 5 x 7

revolutions
-------------
140

Which means revolutions = 140. QED.

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