What is greatest common factor (gcf)?

Why it matters

The GCF is the tool for simplifying fractions: divide the numerator and denominator by their GCF to reach lowest terms in one step. It also helps when you need to split something into the largest equal groups possible — like arranging 12 chairs and 18 desks into the most rows with the same count in each row. It pairs naturally with prime-and-composite-numbers and sets up work with fractions in every grade from 4 through 8.

Worked example

Find the GCF of 12 and 18, then use it to simplify 12/18.

1. 1

   List all the factors of 12 (every whole number that divides 12 evenly): 1, 2, 3, 4, 6, 12.

   12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, 12 ÷ 12 = 1. All give whole-number quotients.

2. 2

   List all the factors of 18: 1, 2, 3, 6, 9, 18.

   18 ÷ 1 = 18, 18 ÷ 2 = 9, 18 ÷ 3 = 6, 18 ÷ 6 = 3, 18 ÷ 9 = 2, 18 ÷ 18 = 1.

3. 3

   Find the factors that appear on both lists: 1, 2, 3, 6.

   These are the "common" factors — numbers that divide into both 12 and 18 without a remainder.

4. 4

   The GREATEST of those shared factors is 6. GCF(12, 18) = 6.

5. 5

   Use the GCF to simplify 12/18: divide the numerator and denominator each by 6. 12 ÷ 6 = 2, 18 ÷ 6 = 3. So 12/18 = 2/3.

   Dividing top and bottom by the same number keeps the fraction's value the same. Using the GCF gets you to fully simplified form in one shot.

6. 6

   Check: 2 and 3 share no common factor other than 1, so 2/3 is in simplest form. ✓

Common mistakes

• •Confusing GCF with LCM. The GCF is the biggest number that fits INTO both numbers (it is smaller than or equal to the smaller number). The LCM is the smallest number that both fit INTO (it is larger than or equal to the larger number).
• •Listing multiples instead of factors — writing "12, 24, 36" when asked for the factors of 12. Factors are always smaller than or equal to the number itself; multiples keep growing.
• •Finding A common factor but not the GREATEST one. The GCF of 12 and 18 is 6, not 2 or 3. Using a smaller factor still simplifies the fraction, just not all the way in one step.
• •Stopping the factor list too early and missing a larger shared factor. For numbers like 24 and 36, a quick pass might stop at 4 and miss that 12 is also a common factor — and is the GCF.

How Briveli teaches greatest common factor (gcf)

Briveli introduces GCF in Grade 4 and 5 through the simplify-fractions routine, then builds to the prime-factorization method for finding GCF with larger numbers in Grade 6, and applies it again in the ratios unit when reducing rates to unit form.