🔢 MathGrade 6Lesson 5
← Back🔒 PDF with full access

Least Common Multiple and Greatest Common Factor

Lesson preview: the Mini Lesson and Vocabulary are open to everyone. Guided practice, exercises, the parent checklist and the printable PDF worksheet come with full access (free 7-day trial or membership).

Learning Objectives

  • 1Find the least common multiple (LCM) of two whole numbers by listing multiples
  • 2Find the greatest common factor (GCF) of two whole numbers by listing factors
  • 3Use prime factorization to compute the LCM and GCF of larger pairs
  • 4Apply LCM and GCF to solve simple real-world word problems
📘

Mini Lesson

When two numbers share factors and multiples, two special values stand out. The Least Common Multiple (LCM) is the smallest non-zero number both can divide into. The Greatest Common Factor (GCF) is the largest number that divides both evenly. LCM helps with adding fractions and scheduling repeated events; GCF helps with simplifying fractions and splitting groups fairly.

Quick Recap: Factors vs Multiples

  • Factors of a number divide INTO it with no remainder. Factors of 12: 1, 2, 3, 4, 6, 12.
  • Multiples of a number come from skip-counting it. First multiples of 4: 4, 8, 12, 16, 20, 24.
  • A "common factor" is a factor shared by two numbers; a "common multiple" is a multiple shared by two numbers.

Least Common Multiple (LCM)

  • Definition: the SMALLEST positive number that is a multiple of BOTH numbers.
  • Method 1 — list multiples of each number until a match appears.
  • Example: multiples of 4 are 4, 8, 12, 16, 20, 24, ... Multiples of 6 are 6, 12, 18, 24, ... The smallest match is 12, so LCM(4, 6) = 12.
  • Method 2 — prime factorization. Take the HIGHEST power of every prime that appears in either number, then multiply.
  • Example: 8 = 2³ and 10 = 2 × 5, so LCM(8, 10) = 2³ × 5 = 40.

Greatest Common Factor (GCF)

  • Definition: the LARGEST whole number that divides BOTH numbers evenly.
  • Method 1 — list factors of each number; the biggest number on both lists is the GCF.
  • Example: factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. The largest shared factor is 6, so GCF(12, 18) = 6.
  • Method 2 — prime factorization. Take the LOWEST power of every SHARED prime, then multiply.
  • Example: 18 = 2 × 3² and 24 = 2³ × 3, so GCF(18, 24) = 2¹ × 3¹ = 6.

When to Use Which

  • Use LCM for "smallest TOTAL" or "next time both happen together" problems — common denominators, repeating schedules.
  • Use GCF for "largest EQUAL group" or "biggest SHARED" problems — simplifying fractions, splitting items evenly.
  • If two numbers share no factor besides 1, their GCF is 1 — they are called RELATIVELY PRIME (e.g. 8 and 15).
  • For any two numbers a and b, the relationship LCM(a, b) × GCF(a, b) = a × b always holds.
LCM of 4 and 6 equals 12Two columns of multiples: 4, 8, 12, 16, 20, 24 and 6, 12, 18, 24, 30, 36, with 12 highlighted as the smallest number on both lists.LCM(4, 6) = 12Multiples of 4Multiples of 648121620246121824303612 is the SMALLEST number on both lists.That makes 12 the Least Common Multiple of 4 and 6.
List multiples until you hit the first match — that is the LCM.
GCF of 12 and 18 equals 6Two columns of factors: 1, 2, 3, 4, 6, 12 and 1, 2, 3, 6, 9, 18, with 6 highlighted as the largest number on both lists.GCF(12, 18) = 6Factors of 12Factors of 18123461212369186 is the LARGEST number on both lists.That makes 6 the Greatest Common Factor of 12 and 18.
List factors of both — the biggest shared value is the GCF.
💡

Tip: LCM is always greater than or equal to the larger of the two numbers, and GCF is always less than or equal to the smaller one. If your answer breaks that rule, recheck.

Vocabulary
LCM
Least Common Multiple — the smallest positive number that is a multiple of both given numbers.
GCF
Greatest Common Factor — the largest whole number that divides both given numbers evenly.
common factor
A whole number that divides each of two or more numbers without a remainder.
common multiple
A number that is a multiple of two or more given numbers at the same time.
prime factorization
A composite number written as a product of prime numbers only, often using exponents.
Free · 7 days · No credit card

Unlock the full HSEA system for 7 days.

Add your first kid profile and your 7 days start: every lesson in every grade, saved progress for each child. Nothing to set up, no card required, nothing to cancel.

  • ✓ All 1,680 lessons, Grades 1–6, every subject
  • ✓ Profiles + saved progress for every kid
  • ✓ Printable lessons with answer key
Start your 7-day free trial →
Already convinced? See full membership plans →
← Lesson 4: Factors, Multiples and Prime FactorizationLesson 6: Powers and Exponents (Intro) →