340 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 21 that is an integer after division by 17, 12 and 10.

Thus:

21 21 21

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

17 12 10

An easy way to do this would be to multiply by 17 x 12 x 10 = 2,040 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 - 21 = 3 x 7

Cog 2 - 17 = 17

Cog 3 - 12 = 2 x 2 x 3

Cog 4 - 10 = 2 x 5

Using these prime factors we can write the fractions as:

3 x 7 and 3 x 7 and 3 x 7

----- --------- ----- and these need to be integers

17 2 x 2 x 3 2 x 5

We can now simplify one of the fractions to give:

3 x 7 and 7 and 3 x 7

----- ----- ----- and these need to be integers

17 2 x 2 2 x 5

To remove the 17 on the first fraction we can multiply throughout by 17 to give:

3 x 7 and 7 x 17 and 3 x 7 x 17

------ ---------- and these need to be integers

2 x 2 2 x 5

We can now ignore the first fraction as it is an integer.

To remove the 2 x 2 on the second fraction we can multiply throughout by 2 x 2 to give:

7 x 17 and 2 x 2 x 3 x 7 x 17

------------------ and these need to be integers

2 x 5

We can ignore the second fraction now as it is an integer, and then simplify the third fraction to give:

2 x 3 x 7 x 17 and this needs to be an integer

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

5

We now finally multiply by 5 to make the last fraction an integer.

We have therefore multiplied by 17, 4, and 5.

17 x 4 x 5 = 340 revolutions. As required.

The easy way from above of 2,040 is exactly 6 times this answer.