What is the trick to finding the greatest common factor of three or more numbers?

Answer 1

The standard ways of determining the greatest common factor of three or more numbers is

* to compare all their factors and choose the largest one that is common to all, or * to compare their prime factors and multiply together all the ones that are common to all, or * to determine the greatest common factor two numbers at a time.

However, you can also look at the difference between the numbers being compared. The greatest common factor cannot be greater than the smallest difference between the numbers. Also, the greatest common factor must be a factor of the difference between the numbers.

Example: Find the greatest common factor of 8, 10, and 14.

The difference between 8 and 10 is 2, which means the greatest common factor cannot be greater than 2. (The difference between 10 and 14 is 4 and the difference between 8 and 14 is 6, both of which are divisible by 2, so 2 remains a possible greatest common factor.) So, check whether 2 evenly divides 8, 10, and 14. It does, so 2 is the greatest common factor.

Example: Find the greatest common factor of 84, 91, and 105.

The difference between 84 and 91 is 7, which means the greatest common factor cannot be greater than 7. (The difference between 91 and 105 is 14 and the difference between 105 and 84 is 21, both of which are divisible by 7, so 7 remains a possible greatest common factor.) So, check whether 7 evenly divides 84, 91, and 105. It does, so 7 is the greatest common factor. Here are the factors for those numbers so you can see that it is correct:

The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

The factors of 91 are 1, 7, 13, and 91.

The factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105.

More Complex Example: Find the greatest common factor of 42, 54, and 66.

The difference between 42 and 54 is 12, which is also the same difference between 54 and 66, so the greatest common factor cannot be greater than 12. So, check whether 12 evenly divides 42, 54, and 66. It does not, so try the largest factor of 12 that is less than 12, which is 6. Check whether 6 divides evenly 42, 54, and 66. It does, so 6 is the greatest common factor. Here are the factors for those numbers so you can see that it is correct:

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

The factors of 66 are 1, 2, 3, 6, 11, 22, 33, and 66.

More Complex Example: Find the greatest common factor of 36, 42, and 76.

The difference between 36 and 42 is 8. The difference between 42 and 76 is 34. Since the greatest common factor must be a factor of the difference, we can examine these smaller numbers, 8 and 34, to see if they have a factor in common. The only factor they have in common (excluding 1) is 2. Since 2 evenly divides 36, 42, and 76, it is the greatest common factor. Here are the factors for those numbers so you can see that it is correct:

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

The factors of 76 are 1, 2, 4, 19, 38, and 76.

Four Numbers Example: Find the greatest common factor of 27, 39, 54, and 60.

The difference between 27 and 39 is 12. The difference between 39 and 54 is 15. The difference between 54 and 60 is 6. Since the greatest common factor must be a factor of the difference, we can examine these smaller numbers, 12, 15, and 6, to see if they have a factor in common. The only factor they have in common (excluding 1) is 3. Since 3 evenly divides 27, 39, 54, and 60, it is the greatest common factor. Here are the factors for those numbers so you can see that it is correct:

The factors of 27 are 1, 3, 9, and 27.

The factors of 39 are 1, 3, 13, and 39.

The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
The same way as with two. You can list the factors and find the common ones or compare their prime factorizations.