260 revolutions.

There are a number of ways of thinking about the solution, and we find this one the quickest way to find the answer.

The total number of teeth moved by Cog 1 will be wholly divisible by each cog in turn, therefore:

Revolutions x Cog 1 ÷ Cog2 is an integer

Revolutions x Cog 1 ÷ Cog3 is an integer

Revolutions x Cog 1 ÷ Cog4 is an integer

So we are after the first number of revolutions x 81 that is an integer after division by 52, 36 and 20.

Thus:

81 81 81

-- and -- and -- all need to be integers (and not fractions).

52 36 20

An easy way to do this would be to multiply by 52 x 36 x 20 = 37,440 revolutions, which would be a correct answer, but not necessarily the smallest answer.

A better way is to break each cog down into its prime factors, where Cog 1 has the largest number of teeth:

Cog 1 - 81 = 3 x 3 x 3 x 3

Cog 2 - 52 = 2 x 2 x 13

Cog 3 - 36 = 2 x 2 x 3 x 3

Cog 4 - 20 = 2 x 2 x 5

3 x 3 x 3 x 3 and 3 x 3 x 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers

------------- ------------- ------------

2 x 2 x 13 2 x 2 x 3 x 3 2 x 2 x 5

Simplifying any fraction that can be simplified gives:

3 x 3 x 3 x 3 and 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers

------------- ----- -------------

2 x 2 x 13 2 x 2 2 x 2 x 5

Multiplying throughout by 2 x 2 gives:

3 x 3 x 3 x 3 and 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers

------------- -------------

13 5

Multiplying throughout 13 gives:

3 x 3 x 3 x 3 and 3 x 3 x 13 and 3 x 3 x 3 x 3 x 13 and these need to be integers

------------------

5

Multiplying throughout 5 gives:

3 x 3 x 3 x 3 x 5 and 3 x 3 x 13 x 5 and 3 x 3 x 3 x 3 x 13

They are all now integers, and we have therefore multiplied by 2 x 2 x 13 x 5.

2 x 2 x 13 x 5 = 260 revolutions. As required.

The easy way from above of 37,440 is exactly 144 times this answer.