Answer: 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.
