Greatest Common Factors (GCF)

Elementary SchoolMiddle School

Definition

A greatest common factor is the largest integer that can completely divide a set of numbers. In other words, GCF is the largest common factor that these numbers share. This can be done by listing out the common factors for a set of numbers and then identifying the number with the highest value amongst them.

Worked examples

\(\)GCF\((12, 18)\): factors of 12 are 1, 2, 3, 4, 6, 12; factors of 18 are 1, 2, 3, 6, 9, 18; \(\)GCF\( = 6\)

List all factors of each number, then pick the largest one that appears in both lists.

\(\)GCF\((24, 36, 48) = 12\) because \(24 = 12 \times 2\), \(36 = 12 \times 3\), \(48 = 12 \times 4\)

The GCF of three or more numbers is the largest integer that divides all of them evenly.

Common mistakes

\(\)GCF\((12, 18) = 36\) → \(\)GCF\((12, 18) = 6\) The GCF must divide both numbers; 36 is a multiple, not a factor. GCF is always ≤ the smallest number in the set.
\(\)GCF\((8, 12) = 4\) so list only common factors: 4 → List all common factors: 1, 2, 4, then pick the greatest Students sometimes skip listing 1 and 2, missing the systematic check for the largest common factor.
\(\)GCF\((15, 28) = 5\) → \(\)GCF\((15, 28) = 1\) 5 divides 15 but not 28. When numbers share no prime factors, their GCF is 1 (they are coprime).

Where you'll use it next

You'll use GCF to simplify fractions to lowest terms, solve problems involving repeated patterns or grouping, and find common denominators. It's foundational for factoring polynomials and working with rational expressions in algebra.

Found in 1 StudyPug lesson

Mastering Greatest Common Factors (GCF)

8th Grade Math

Unlock the power of GCF to simplify fractions, solve equations, and boost your math skills. Learn efficient methods like prime factorization to tackle complex problems with confidence.

8th Grade8th