Key takeaway: The greatest common factor (GCF) of two numbers is the largest whole number that divides both without a remainder, such as 36 for 72 and 108.
Grade 66.NS.B.4Numbers and Operations
Definition of Greatest Common Factor
The greatest common factor, or GCF, is the biggest number that is a factor of every number in a group. Both 72 and 108 can be split evenly by 4, by 12 and by 18, but the top of that list is 36.
Listing every factor works for small numbers. For bigger ones, break each number into primes and multiply only the primes that appear in both. Reducing fractions and pulling a common piece out of a sum both rely on the GCF.
Formula and notation
The GCF of two numbers is written
\[\gcd(72,\,108) = 36 \qquad\text{or}\qquad \text{GCF}(72, 108) = 36\]
Pulling out the GCF turns a sum into a product: \(72 + 108 = 36 \times (2 + 3)\).
Key facts
- The greatest common factor is never larger than the smallest of the numbers.
- Two numbers whose GCF is 1 are called relatively prime.
- If one number divides the other, the smaller number is the GCF.
- Dividing both terms of a fraction by their GCF puts it in lowest terms.
Greatest Common Factor example
A coach has 72 granola bars and 108 juice boxes. She wants identical snack packs that use every item, with as many packs as possible. How many packs can she build, and what does each pack hold?
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Factor: \(72 = 2^3 \times 3^2\) and \(108 = 2^2 \times 3^3\).
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Keep the lower power of each shared prime: \(2^2 \times 3^2 = 36\).
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Per pack: \(72 : 36 = 2\) bars, and \(108 : 36 = 3\) boxes.
She builds 36 packs, two bars and three boxes in every one.
Common mistake
Watch out
Zoe sees that 12 divides both 72 and 108 and writes that their greatest common factor is 12.
Why it is wrong
Twelve is a shared factor, but not the greatest one. Dividing by 12 leaves 6 and 9, which still share the factor 3. Multiplying \(12 \times 3\) gives the true GCF, 36. Keep dividing until the remaining numbers have no common factor above 1.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations