Math Core

You have 48 pencils and 36 erasers, and you want to make identical supply kits with nothing left over. What is the greatest number of kits you can make? The answer is a number that divides both 48 and 36, and it has to be the biggest one. That number is called the greatest common factor.

Factors and common factors

A factor of a number divides it with no remainder. The factors of 18 are 1, 2, 3, 6, 9 and 18, because each one divides 18 evenly.

To find all the factors of a number, look for factor pairs. Start at 1 and work up, stopping when the pairs start to repeat:

18 = 1 \times 18 = 2 \times 9 = 3 \times 6

A common factor of two numbers is a factor of both of them.

Definition

Greatest common factor (GCF)

The greatest common factor of two or more whole numbers is the largest number that is a factor of all of them.

Method 1: List the factors

List the factors of each number, circle the ones they share, and pick the largest.

Worked example: Listing factors

Find the GCF of 18 and 30.

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

The common factors are 1, 2, 3 and 6. The greatest is 6.

Method 2: Use prime factorization

Listing works well for small numbers. For bigger numbers, break each number into prime factors. A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, and so on.

To find the prime factorization, keep dividing by primes until only primes are left. For example, $84 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 7$ .

GCF from prime factorizations

Write each number as a product of primes. The GCF is the product of the primes the numbers share, using each shared prime as many times as it appears in both numbers.

Worked example: Prime factorization

Find the GCF of 84 and 60.

\begin{aligned} 84 &= 2 \times 2 \times 3 \times 7 \\ 60 &= 2 \times 2 \times 3 \times 5 \end{aligned}

Both numbers share two 2s and one 3. The 7 and the 5 are not shared. So the GCF is $2 \times 2 \times 3 = 12$ .

Check: $84 \div 12 = 7$ and $60 \div 12 = 5$ . Both divide evenly. ✓

Tip

To check a GCF, divide both numbers by it. If the two results have no common factor except 1 (like 7 and 5 above), you found the greatest one. If they still share a factor, your GCF is too small.

Factoring out the GCF

The GCF lets you rewrite a sum as a product using the distributive property. Since $42 = 14 \times 3$ and $70 = 14 \times 5$ :

42 + 70 = 14 \times 3 + 14 \times 5 = 14(3 + 5)

Both sides equal 112. Pulling out the GCF like this is called factoring, and you'll use it a lot in algebra.

Worked example: Rewrite a sum with the GCF

Use the GCF to rewrite $36 + 60$ as a product.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest of these that also divides 60 is 12. So the GCF is 12.

$36 = 12 \times 3$ and $60 = 12 \times 5$ , so $36 + 60 = 12(3 + 5)$ .

Check: $12 \times 8 = 96$ and $36 + 60 = 96$ . ✓

Common mistake

Using a common factor that isn't the greatest is a common slip. $36 + 60 = 6(6 + 10)$ is true, but 6 isn't the GCF, because 6 and 10 still share a factor of 2. When a problem asks for the GCF, make sure the numbers left inside the parentheses share no factor except 1.

GCF in word problems

Look for the GCF when you split two groups into identical sets with nothing left over and want the greatest number of sets (or the largest possible size).

Worked example: Supply kits

You have 48 pencils and 36 erasers. What is the greatest number of identical kits you can make with none left over? What goes in each kit?

The number of kits must divide both 48 and 36, and you want the largest such number.

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The GCF is 12. You can make 12 kits. Each kit has $48 \div 12 = 4$ pencils and $36 \div 12 = 3$ erasers.