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What is the
Greatest Common Factor? 

The greatest common factor (GCF) is a concept that describes the relationship between 2 or more numbers. 

The factors of a number are all of the numbers that can be divided evenly into it. For example, the factors of 6 are 1, 2, 3, and 6 because 6 can be divided evenly by each of those numbers. 

The greatest common factor of a pair of numbers is the largest number that both of the numbers can be evenly divided by. 

How to Find the Greatest Common Factor

Sometimes, you can calculate the GCF in your head by asking yourself the question, "What is the largest possible number that both of these numbers can be divided by?"

Example: Find the GCF of 12 & 14

What number can both 12 and 14 be divided by?

Well, they can both be divided by 2.

Is that the largest number they can both be divided by?

Yes, so the GCF of 12 & 14 is 2.

Example: Find the GCF of 3 & 15

What number can both 3 and 15 be divided by?

They can both be divided by 3.

Is that the largest number they can both be divided by?

Yes, so the GCF of 3 & 15 is 3.

If the numbers are small, it's pretty easy to calculate the GCF in your head. However, if the numbers are bigger, I recommend using the prime factorization method.

To find the GCF using prime factorization...

  1. List the prime factors of each number.
  2. Find the factors that the numbers have in common.
  3. Multiply all the common factors to find the GCF. 

Example: Find the GCF of 48 & 108

Step 1: List the prime factors of each number. 

\(48=2\times2\times2\times2\times3\)

\(108 = 2\times2\times3\times3\times3\)

Step 2: Find the factors that the numbers have in common.

\(48=\color{black}{2}\times\color{black}{2}\times2\times2\times\color{black}{3}\)

\(108 = \color{black}{2}\times\color{black}{2}\times\color{black}{3}\times3\times3\)

Step 3: Multiply all the common factors to find the GCF. 

\(\color{black}{2\times2\times3} = 12\)

Answer: The greatest common factor of 48 and 108 is 12.

Finding the GCF of Three or More Numbers

Usually, you will only be asked to find the greatest common factor of a pair of numbers. 

However, if you are ever asked to find the GCF of three or more numbers, the process is basically the same.

You just have to make sure that you only multiply the factors that ALL of the numbers have in common. 

Example: Find the GCF of 60, 140, & 450

Step 1: List the prime factors of each number. 

\(60=2\times2\times3\times5\)

\(140 = 2\times2\times5\times7\)

\(450 = 2\times3\times3\times5\times5\)

Step 2: Find the factors that ALL the numbers have in common. 

\(60=\color{black}{2}\times2\times3\times\color{black}{5}\)

\(140 = \color{black}{2}\times2\times\color{black}{5}\times7\)

\(450 = \color{black}{2}\times3\times3\times\color{black}{5}\times5\)

In this example, 60 and 450 share a factor of 3, but it doesn't count as a common factor because 140 doesn't have a factor of 3.

Similarly, the second 2 that 60 and 140 share doesn't count as a common factor because 450 doesn't have a second 2 as a factor. 

Step 3: Multiply all the common factors to find the GCF. 

\(\color{black}{2\times5} = 10\)

Answer: The greatest common factor of 60, 140, and 450 is 10.

Using the GCF to Reduce Fractions

The greatest common factor is an awesome tool to use when you are reducing fractions. 

To reduce fractions, you have to divide the numerator and denominator by the same number.

You can reduce fractions by dividing the fraction multiple times like this:

Reduce \(\frac{30}{45}\)

Step 1: 30 and 45 are both divisible by 5.

\(\frac{30}{45} {\div5 \atop \div5} = \frac{6}{9}\)

Step 2: 6 and 9 are both divisible by 3. 

\(\frac{6}{9}{\div3\atop\div3} =\frac{2}{3}\)

Answer: 2 and 3 are relatively prime, so \(\frac{2}{3}\) is the completely reduced form of \(\frac{30}{45}\).

However, reducing fractions can be much easier if you use the greatest common factor.

The GCF of the numerator and the denominator is the largest number that they can both be divided by.

So, if you divide by the GCF, you will only have to reduce once instead of multiple times.

Reduce \(\frac{30}{45}\)

Step 1: The GCF of 30 and 45 is 15.

\(\frac{30}{45} {\div15 \atop \div15} = \frac{2}{3}\)

Answer: 2 and 3 are relatively prime, so \(\frac{2}{3}\) is the completely reduced form of \(\frac{30}{45}\).

Using the GCF to Find the LCM

The greatest common factor can also be used to find the least common multiple of a pair of numbers. 

Visit this page to learn more about how to use the GCF to find the LCM. 

How to Find the GCF of Polynomials

In addition to finding the GCF of normal numbers, you can also find the greatest common factor of unknown numbers (variables) in polynomials. 

The most common application of this is finding the greatest common factor of polynomials in order to simplify the polynomials into factored form. 

Visit this page to learn more about finding the greatest common factor of polynomials.

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