What is the trick to finding the greatest common factor?

The following answer describes four methods of finding the greatest common factor, with examples, and several "tricks" or shortcuts that can make it easier.

Method: Guess and Refine

Sometimes, you can look at two numbers and make a good guess that you can refine.

Example 1: Find the greatest common factor of 45 and 50.

Because both numbers end in either a 5 or 0, you know that they are both divisible by 5. If you divide both numbers by 5 and the results have no common factors (except 1), 5 is the greatest common factor.

45 ÷ 5 = 9

50 ÷ 5 = 10

Since 9 and 10 are consecutive numbers, they have no common factors. Therefore, the greatest common factor is 5.

Example 2: Find the greatest common factor of 150 and 750.

Both numbers end in 50, so they are both divisible by 50. If you divide both numbers by 50 and the results have another common factor, you continue identifying common factors until you have a pair without common factors.

150 ÷ 50 = 3

750 ÷ 50 = 15

Since 15 is divisible by 3, and 3 is divisible by 3, you have another common factor, which is 3. Then, you can divide the most recent results by 3.

3 ÷ 3 = 1

15 ÷ 3 = 5

Since 1 and 5 do not have any common factors, take the two factors that you did identify, 50 and 3, and multiply them together: 50 x 3 = 150. This number, 150, is the greatest common factor.

Method: Find All the Factors

If the numbers are small enough or you know that they have only a few factors, you can list all the factors of each number and compare to determine the largest factor they have in common. One of the related questions links will take you to a page with the complete list of factors for numbers 1 through 100.

Example: Find the greatest common factor of 15 and 18.

The factors of 15 are 1, 3, 5, and 15.

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

The common factors are 1 and 3, so the greatest common factor is 3.

Example: Find the greatest common factor of 26 and 91.

The factors of 26 are 1, 2, 13, and 26.

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

The common factors are 1 and 13, so the greatest common factor is 13.

Method: Find the Prime Factors

In situations where you cannot get a good start simply by looking at the numbers, follow the following steps:

1. Determine the prime factors of each number. See the related question "How do you find prime factors" for a method on doing this. Also, one of the related questions links will take you to a page with the complete list of prime factors for numbers 1 through 100.

2. Determine the prime factors they have in common.

3. Multiply all the prime factors they have in common to calculate the greatest common factor. Example: Find the greatest common factor of 5,544 and 37,620.

The prime factors of 5,544 are 2, 2, 2, 3, 3, 7, and 11.

The prime factors of 37,620 are 2, 2, 3, 3, 5, 11, and 19.

The common prime factors are 2, 2, 3, 3, and 11.

Therefore, the greatest common factor is 2 x 2 x 3 x 3 x 11 = 396. Example: Find the greatest common factor of 7,888 and 10,002.

The prime factors of 7,888 are 2, 2, 2, 2, 17, and 29.

The prime factors of 10,002 are 2, 3, and 1667.

The common prime factors are a single 2.

Therefore, the greatest common factor is 2. Method: Euclidean Algorithm

This method is more efficient than finding the prime factors when the numbers are large, but teachers might prefer that you gain experience determining the prime factors of numbers. For this method, divide the larger number by the smaller number, then divide the "divisor" from the previous division by the remainder from the previous division, and continue until a number divides evenly. That divisor is the greatest common factor. Example: Find the greatest common factor of 33 and 77.

77 ÷ 33 = 2 remainder 11

33 ÷ 11 = 3 with no remainder

So, the final divisor, 11, is the greatest common factor. Example: Find the greatest common factor of 27 and 168.

168 ÷ 27 = 6 remainder 6

27 ÷ 6 = 4 remainder 3

6 ÷ 3 = 2 with no remainder

So, the final divisor, 3, is the greatest common factor.

---- Shortcut 1: If one number is a multiple of the other, the smaller number is the greatest common factor, because it is the largest possible factor of itself.

Example: Find the greatest common factor of 72 and 288.

288 is divisible by 72, therefore 72 is the greatest common factor.

Shortcut 2: The greatest common factor of two numbers cannot be larger than the difference between the two numbers. So, you only need to test the numbers that are equal to or less than the difference between those two numbers. Also, the greatest common factor must be a factor of the difference between the two numbers. (This shortcut can help with finding the greatest common factor of three or more numbers. Examples are shown in the related question on finding the greatest common factor of three or more numbers.)

Example: Find the greatest common factor of 56 and 64.

The difference between 56 and 64 is 64 - 56 = 8. The largest possible common factor is the difference itself. So, check whether 8 divides evenly into both of them.

56 ÷ 8 = 7

64 ÷ 8 = 8

Therefore, 8 is the greatest common factor. Example: Find the greatest common factor of 72 and 88.

The difference between 88 and 72 is 88 - 72 = 16. Check whether 16 divides evenly into both of them. It does not. But, the greatest common factor must be a factor of 16. The factors of 16 are 1, 2, 4, 8, and 16. So, try the next largest factor, 8, and see if it divides evenly into both of them.

72 ÷ 8 = 9

88 ÷ 8 = 11

Therefore, 8 is the greatest common factor.

Example: Find the greatest common factor of 1003 and 1180.

The difference between 1180 and 1003 is 177. Check whether 177 divides evenly into both of them. It does not. But, the greatest common factor must be a factor of 177. By using the divisibility rule for 3, you know that 3 is a factor of 177, but the divisibility rule indicates that neither 1003 nor 1180 are divisible by 3. 177 ÷ 3 = 59, so check 59 as a factor of both numbers. Note that 3 and 59 are both prime numbers, so they are the only prime factors of 177, so if there is a greatest common factor of 1003 and 1180 other than 1, since we have ruled out 177 and 3, it must be 59.

1003 ÷ 59 = 17

1180 ÷ 59 = 20

Therefore, 59 is the greatest common factor. Corollary 1 to Shortcut 2: If the numbers are only one number apart, they are relatively prime and have no common factor other than 1. Example: Find the greatest common factor of 4 and 5.

The difference is 1, so the greatest common factor is 1. They are relatively prime.

Corollary 2 to Shortcut 2: If the difference between the two numbers is 2 and the numbers are not even numbers, they are relatively prime and have no common factor other than 1. If the difference is 2 and they are both even, the greatest common factor is 2.

Example: Find the greatest common factor of 13 and 15.

The difference is 2 and the numbers are not even, so the greatest common factor is 1. Example: Find the greatest common factor of 14 and 16.

The difference is 2 and the numbers are even, so the greatest common factor is 2.

Corollary 3 to Shortcut 2: If the difference between the two numbers is a prime number, either that number is the greatest common factor or 1 is the greatest common factor. Example: Find the greatest common factor of 40 and 69.

The difference is 29, which is a prime number. Since 29 does not divide evenly into both 40 and 69, the greatest common factor is 1, which means they are relatively prime. Example: Find the greatest common factor of 91 and 104.

The difference is 13, which is a prime number. Since 13 divides evenly into both 91 and 104, the greatest common factor is 13.

91 ÷ 13 = 7

104 ÷ 13 = 8 Shortcut 3: If one of the numbers is prime, either it is the greatest common factor or the greatest common factor is 1. (Its only factors are 1 and itself, so those are the only possible common factors it could have with another number.)

Example: Find the greatest common factor of 83 and 90.

83 is a prime number and it is not a factor of 90, so the greatest common factor is 1. Example: Find the greatest common factor of 41 and 246.

41 is a prime number and it is a factor of 246, so the greatest common factor is 41.

246 ÷ 41 is 6

---- Divisibility Rules:

To determine the prime factors, it is sometimes helpful to use the divisibility rules.

2: The number ends in 0, 2, 4, 6, or 8.

Examples: 14, 58, 100, 3336

3: The sum of the number's digits is divisible by 3.

Examples: 78 (7+8=15 which is divisible by 3), 114 (1+1+4=6 which is divisible by 3)

5: The number ends in 0 or 5.

Examples: 70, 195, 4860

7: The last digit doubled subtracted from the rest of the number is divisible by 7 or is equal to 0.

Examples: 343 (3x2=6; 34-6=28 which is divisible by 7), 875 (5x2=10; 87-10=77 which is divisible by 7)

11: Start with the left-most digit, subtract the next one, add the next one, subtract the next one, etc., and the final result is divisible by 11 or is equal to 0.

Examples: 165 (1-6+5=0), 308 (3-0+8=11 which is divisible by 11), 1078 (1-0+7-8=0)

Prime Numbers: Prime factors are prime numbers. The first 25 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.