6 — Greatest Common Factor

Greatest Common Factor

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The Greatest Common Factor (GCF) is the biggest number that divides evenly into two or more numbers. Think of it like finding the largest group size when you want to share things equally! For example, if you have 12 stickers and 8 pencils, the GCF tells you the most groups you can make where each group has the same number of stickers and pencils.

Tip

The GCF is never bigger than the smallest number you're working with. If your answer is larger than the smallest number, check your work!

Greatest Common Factor (GCF) · 1:19

Worked Example

Problem

Find the GCF of 16 and 20 using the listing method.

Solution

4

Explanation

  1. 1List all factors of 16: 1, 2, 4, 8, 16.
  2. 2List all factors of 20: 1, 2, 4, 5, 10, 20.
  3. 3Circle the common factors: 1, 2, 4.
  4. 4The greatest common factor is 4.

Worked Example

Problem

Find the GCF of 30 and 42 using prime factorization.

Solution

6

Explanation

  1. 1Find prime factorization of 30: 30 = 2 × 3 × 5.
  2. 2Find prime factorization of 42: 42 = 2 × 3 × 7.
  3. 3Circle the common prime factors: 2 and 3.
  4. 4Multiply the common factors: 2 × 3 = 6. The GCF is 6.

Question

Find the GCF of 6 and 9.

Show Answer
3
Show Solution
  1. 1List factors of 6: 1, 2, 3, 6.
  2. 2List factors of 9: 1, 3, 9.
  3. 3Find common factors: 1, 3.
  4. 4The greatest common factor is 3.

Question

Find the GCF of 8 and 12.

Show Answer
4
Show Solution
  1. 1List factors of 8: 1, 2, 4, 8.
  2. 2List factors of 12: 1, 2, 3, 4, 6, 12.
  3. 3Find common factors: 1, 2, 4.
  4. 4The greatest common factor is 4.

Question

Find the GCF of 10 and 15.

Show Answer
5
Show Solution
  1. 1List factors of 10: 1, 2, 5, 10.
  2. 2List factors of 15: 1, 3, 5, 15.
  3. 3Find common factors: 1, 5.
  4. 4The greatest common factor is 5.

Question

Find the GCF of 20 and 30.

Show Answer
10
Show Solution
  1. 1List factors of 20: 1, 2, 4, 5, 10, 20.
  2. 2List factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
  3. 3Find common factors: 1, 2, 5, 10.
  4. 4The greatest common factor is 10.

Question

Find the GCF of 18, 24, and 30.

Show Answer
6
Show Solution
  1. 1List factors of 18: 1, 2, 3, 6, 9, 18.
  2. 2List factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  3. 3List factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
  4. 4Find common factors of all three: 1, 2, 3, 6.
  5. 5The greatest common factor is 6.

Question

Use prime factorization to find the GCF of 48 and 72.

Show Answer
24
Show Solution
  1. 1Find prime factorization of 48: 48 = 2⁴ × 3 = 2 × 2 × 2 × 2 × 3.
  2. 2Find prime factorization of 72: 72 = 2³ × 3² = 2 × 2 × 2 × 3 × 3.
  3. 3Find common prime factors: 2³ × 3¹ (use the lowest power of each common prime).
  4. 4Multiply: 2³ × 3 = 8 × 3 = 24. The GCF is 24.

Question

Find the GCF of 45, 60, and 75.

Show Answer
15
Show Solution
  1. 1Find prime factorization of 45: 45 = 3² × 5.
  2. 2Find prime factorization of 60: 60 = 2² × 3 × 5.
  3. 3Find prime factorization of 75: 75 = 3 × 5².
  4. 4Find common prime factors: 3¹ × 5¹ (use the lowest power of each common prime).
  5. 5Multiply: 3 × 5 = 15. The GCF is 15.

Tip

Remember: The GCF is the largest number that divides evenly into all the numbers you're working with. It's super useful for simplifying fractions and solving problems about sharing things equally!