All skills Math

Learn Index · Number Sense and Numeration Grade 6–8

Greatest common factor

The largest number that divides two numbers exactly.

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What is the GCF?

The greatest common factor (GCF), also called the greatest common divisor, is the largest number that divides two or more numbers with no remainder.

For 12 and 18:

  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Common factors: 1, 2, 3, 6
  • GCF = 6

Method 1: List the factors

List every factor of each number, circle the common ones, pick the biggest. Good for small numbers.

Method 2: Prime factorization

  1. Write each number’s prime factorization.
  2. Find the primes they share.
  3. Multiply the shared primes (using the smaller power of each).

Example: GCF of 60 and 84

  • 60 = 2² × 3 × 5
  • 84 = 2² × 3 × 7
  • Shared: 2² and 3
  • GCF = 4 × 3 = 12

Method 3: Euclidean algorithm

Great for big numbers. Divide the bigger number by the smaller and keep the remainder. Then divide the old divisor by the remainder. Repeat until the remainder is 0. The last non-zero remainder is the GCF.

Example: GCF of 252 and 105

Division Remainder
252 ÷ 105 = 2 42
105 ÷ 42 = 2 21
42 ÷ 21 = 2 0

GCF = 21. The Euclidean algorithm is over 2,000 years old and is still used in computer science.

Using the GCF

Simplifying fractions: Divide top and bottom by the GCF to simplify in one step.

  • 24/36: GCF = 12 → 2/3

Sharing into equal groups: You have 24 pencils and 36 erasers and want identical kits with nothing left over. The most kits is the GCF: 12 kits, each with 2 pencils and 3 erasers.

Factoring expressions: 6x + 9 = 3(2x + 3), where 3 is the GCF.

Special cases

  • If one number divides the other, the smaller one is the GCF: GCF(8, 24) = 8.
  • If two numbers share no prime factors, the GCF is 1. They’re called relatively prime: GCF(8, 15) = 1.

Worked examples

Example 1: GCF of 12 and 18.

6 (shown above).

Example 2: GCF of 45 and 75.

45 = 3² × 5; 75 = 3 × 5². Shared: 3 × 5 = 15.

Example 3: Simplify 42/56.

GCF = 14 → 3/4.

Common mistakes

  • Giving the least common multiple instead.
  • Choosing a common factor that isn’t the greatest, like 2 for 12 and 18.
  • Multiplying all the primes instead of just the shared ones.

Try it yourself

  1. GCF of 16 and 40
  2. GCF of 27 and 36
  3. GCF of 14 and 25
  4. Simplify 18/48 using the GCF.
  5. 30 apples and 45 oranges go into identical bags. What is the most bags you can make?

Answers: 1) 8 2) 9 3) 1 4) 3/8 5) 15

Quick recap

  • The greatest common factor is the biggest shared factor.
  • Find it by listing, prime factorization or the Euclidean algorithm.
  • Use it to simplify fractions and make equal groups.