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Greatest Common Factor and Least Common Multiple

A free Pre-Algebra lesson from the “Number Foundations” unit, with a worked example and practice problems including step-by-step solutions.

The GCF (Greatest Common Factor) is the largest whole number that divides into two numbers evenly. The LCM (Least Common Multiple) is the smallest positive number that both numbers divide into. Both can be found by listing factors or multiples, or by using prime factorization.

What you'll learn

Why it matters: Splitting things into equal groups uses GCF (the largest equal split possible). Scheduling repeating events uses LCM (when two cycles next line up).

Worked example

Problem. Find the GCF and LCM of 12 and 18.

  1. Factors of 12: 1, 2, 3, 4, 6, 12.
  2. Factors of 18: 1, 2, 3, 6, 9, 18. The largest shared factor is 6, so GCF = 6.
  3. Multiples of 12: 12, 24, 36, 48. Multiples of 18: 18, 36, 54. The smallest shared multiple is 36, so LCM = 36.

Answer: GCF 6, LCM 36

Practice problems

1. Find the GCF of 8 and 12.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the GCF of 8 and 12.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Factors of 8: 1, 2, 4, 8.
  4. Factors of 12: 1, 2, 3, 4, 6, 12. Largest shared factor is 4.
  5. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

2. Find the GCF of 15 and 25.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the GCF of 15 and 25.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Factors of 15: 1, 3, 5, 15.
  4. Factors of 25: 1, 5, 25. Largest shared factor is 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

3. Find the LCM of 4 and 6.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the LCM of 4 and 6.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiples of 4: 4, 8, 12, 16.
  4. Multiples of 6: 6, 12, 18. Smallest shared multiple is 12.
  5. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

4. Find the LCM of 3 and 5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the LCM of 3 and 5.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. When two numbers share no factors, the LCM is their product.
  4. 3 x 5 = 15.
  5. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

5. Use the GCF to simplify 14/21.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Use the GCF to simplify 14/21.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. GCF of 14 and 21 is 7.
  4. 14 / 7 = 2 and 21 / 7 = 3, so 14/21 = 2/3.
  5. Check the result by substituting or estimating: the response should match 2/3 and make sense in the original problem.

Answer: 2/3

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