Greatest Common Factor (GCF)

The greatest common factor of a set of numbers is the largest number that can evenly divide all the given numbers. GCF is also called as greatest common divisor or GCD.

There are two ways or methods in determining the greatest common factor: list method and prime factorization method.

A quick warning before we continue, although list method works just fine, I don't think that it is the most efficient method. The reason is that as numbers become larger, list method becomes VERY tedious. I highly recommend to use the Prime Factorization method in finding the GCF.

For this exercise though, it's nice to see how both are utilized. By doing so, you will understand the advantage of one method over the other.

 List Method

 Step 1: List all possible factors or divisors of each number
 Step 2: The greatest common factor is the largest common factor in the list. Be careful, there will be many common factors, but you have to PICK the one that is the greatest in value!

Let's take a look at some examples!

Example 1: Suppose you are to find the GCF of 18 and 60.

 Given numbers List of possible factors or divisors 18 → 1, 2, 3, 6, 9, and 18 60 → 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60

The numbers in blue color are common factors or divisors for 18 and 60, but the one in red is the largest common divisor in the list.

Therefore, the GCF of 18 and 60 is 6. In compact form, GCF (18,60) = 6.

Example 2: Find the GCF of 20 and 24.

 Given numbers List of possible factors or divisor 20 → 1, 2, 4, 5, 10 and 20 24 → 1, 2, 3 ,4, 6, 8, 12, and 24.

The numbers in blue are common divisors but the one in red is the greatest common divisor.

And so, the GCF of 20 and 24 is just 4. In shorthand form, GCF (20,24) = 4.

 Prime Factorization Method

Basically, the key steps behind Prime Factorization Method for GCF is as follows:

 Step 1: Express each number as product of exponential numbers having a prime base. Step 2: Multiply together the exponential numbers with the lowest powers for each "common copy" of the the listed prime numbers.

Example 1: Find the GCF of 12 and 90.

Step 1. Write the prime factors of each number.
 12 = 2 x 2 x 3 = (22 ) (3) 90 = 2 x 3 x 3 x 5 = (2) (32 ) (5)

Step 2. Identify the exponential number with the lowest power for each "common copy" of prime numbers listed.

• Tip: Align the terms having the same base to compare their exponents. For each exponential number with a prime base, pick the one with the lowest power.
• Don't include 5 because it is not a common prime number in both factorizations.

 12 = (22 ) (3) 90 = (2)   (32 )  (5)
Step 3. Multiply together the exponential numbers picked in step 2 to determine the GCF   GCF (12, 90) = 2 x 3 = 6

Example 2: Find the GCF of 100 and 280.

Step 1. Write the prime factors of each number.
 100 = 2 x 2 x 5 x 5 = (22 ) (52) 280 = 2 x 2 x 2 x 5 x 7 = (23) (5) (7)

Step 2. Identify the exponential number with the lowest power for each "common copy" of prime numbers listed.

• Tip: Align the terms having the same base to compare their exponents. For each exponential number with a prime base, pick the one with the lowest power.
• Don't include 7 because it is not a common prime number.

 100 = (22)  (52) 280 = (23)   (5)   (7)
Step 3. Multiply together the exponential numbers picked in step 2 to determine the GCF

GCF (100,280) = 22 x 5 = 4 x 5 = 20

Example 3: Find the GCF of 75, 315 and 420.

Step 1. Write the prime factors of each number.
 75 = 3 x 5 x 5 = (3) (52) 315 = 3 x 3 x 5 x 7 = (32) (5) (7) 420 = 2 x 2 x 3 x 5 x 7 = (22) (3) (5) (7)

Step 2. Identify the exponential number with the lowest power for each "common copy" of prime numbers listed.

• Tip: Align the terms having the same base to compare their exponents. For each exponential number with a prime base, pick the one with the lowest power.
• Don't include 2 and 7 because they are not common prime numbers.

 75 = (3)     (52) 315 = (32)    (5)   (7) 420 = (22)  (3)      (5)   (7)
Step 3. Multiply together the exponential numbers picked in step 2 to determine the GCF   GCF (75, 315, 420) = 3 x 5 = 15

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