Using the Euclidean algorithm, you can find that the greatest common factor of 540 and 132 is 12.
First, you need to see what the remainder is after dividing 540 by 132.
540/132 = 4 R 12
Then, you need to see if 12 goes evenly into 132, or if there is a further remainder.
132/12 = 11 with no remainder.
Since there is no further remainder, 12 is the GCF.
The way you would express this using the Euclidean algorithm is:
540 - (4 x 132) = 12
132 - (11 x 12) = 0
First and foremost do not panic about the size of the bigger number. Since you are in search of the Greatest Common Factor, you are looking for the "math facts" of each of the numbers and you want the largest one that apply to both. Do you know your rules of divisibility? If so, that is of great benefit. If not, you will want to familiarize yourself with them.
Begin with 132.
132 X 1 66 X 2 (it is even so it 2 has to work)
44 X 3 (the sum of the digits is divisible by 3 so it is divisible by 3 -- 1 + 2 + 3= 6)
33 X 4 (it is divisible by 4 because the last two digits are divisible by 4)
22 X 6 (6 has to work because 2 and 3 work)
12 X 11 (this is the last pair of factors because they are right next to each other numerically. This doesn't hold true always but when it does work out that way, it is a great clue to let you know you have all the factors)
Now 540 is our other number. The greatest common factor of 540 and 132 cannot be any larger than 132. I know from the list of 132 that the GCF is larger than 10 because 10 X 54 is 540 and 54 is not a factor of 132. The only two factors left from 132 are 11 and 12. Nothing multiplied by 1 will give me anything ending with 0 (the ones place of 540) so it has to be 12.
Enjoy math. It really can be fun.