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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. In Number Foundations, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

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.
  4. Connect the calculation back to Greatest Common Factor and Least Common Multiple so the method, not just the arithmetic, is clear.

Answer: GCF 6, LCM 36

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Find the GCF of 8 and 12.

Show solution
  1. Factors of 8: 1, 2, 4, 8.
  2. Factors of 12: 1, 2, 3, 4, 6, 12. Largest shared factor is 4.

Answer: 4

2. Find the GCF of 15 and 25.

Show solution
  1. Factors of 15: 1, 3, 5, 15.
  2. Factors of 25: 1, 5, 25. Largest shared factor is 5.

Answer: 5

3. Find the LCM of 4 and 6.

Show solution
  1. Multiples of 4: 4, 8, 12, 16.
  2. Multiples of 6: 6, 12, 18. Smallest shared multiple is 12.

Answer: 12

4. Find the LCM of 3 and 5.

Show solution
  1. When two numbers share no factors, the LCM is their product.
  2. 3 x 5 = 15.

Answer: 15

5. Use the GCF to simplify 14/21.

Show solution
  1. GCF of 14 and 21 is 7.
  2. 14 / 7 = 2 and 21 / 7 = 3, so 14/21 = 2/3.

Answer: 2/3

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