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

The largest cog has 102 teeth and the others have 60, 46 and 34 respectively.

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

We are after a number such that (revolutions * largest cog) / (every small cog) is an integer. This is easily achieved by looking at the cogs' prime factors:

102 = 2 * 3 * 17
60  = 2 * 2 * 3 * 5
46  = 2 * 23
34  = 2 * 17

So

revolutions * 102
-----------------------
2 * 2 * 3 * 5 * 23 * 17

revolutions * 2 * 3 * 17
= ------------------------
2 * 2 * 3 * 5 * 23 * 17

revolutions
= -----------
2 * 5 * 23

Which means revolutions = 230. QED

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