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Mathematics › ⚡ Grade 6 › GCF & LCM: Splitting Bags and Syncing Lights
⚡ Grade 6 · Lesson 7 of 14

GCF & LCM: Splitting Bags and Syncing Lights

The Greatest Common Factor splits equally; the Least Common Multiple is the first sync point.

Grade 6Middle School
GCF & LCM: Splitting Bags and Syncing Lights — illustration
💡
The big idea: GCF (Greatest Common Factor) answers: 'What is the largest group size that divides both numbers evenly?' LCM (Least Common Multiple) answers: 'What is the first number both sequences hit?' These two ideas are mirrors: GCF shrinks toward the shared divisor, LCM grows toward the shared multiple. They are linked by the identity: GCF(a, b) × LCM(a, b) = a × b.
🎯 By the end, you'll be able to
  • Find the GCF of two numbers using factor lists or prime factorisation
  • Find the LCM of two numbers using multiple lists or prime factorisation
  • Apply the identity GCF × LCM = a × b
  • Solve real-world problems using GCF and LCM
📎 You should already know
  • Factors and multiples (Grade 4)
  • Prime numbers and prime factorisation

GCF — splitting bags equally

You have 12 apples and 18 oranges. You want to split them into identical gift bags, with no fruit left over, using the largest possible number of bags. That largest number is the GCF.

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 (the largest shared factor). So 6 bags, each with 2 apples and 3 oranges.

LCM — syncing traffic lights

Traffic light A turns green every 4 minutes; light B turns green every 6 minutes. Starting now (both green), when is the first time both are green at the same time again? That first shared moment is the LCM.

Multiples of 4: 4, 8, 12, 16, … Multiples of 6: 6, 12, 18, … First common multiple: 12. Both lights sync again after 12 minutes.

🎮 GCF & LCM Explorer LIVE
In GCF mode, two dot-rectangles show the factor arrangements for each number — tiles highlighted in both colors are common factors; the largest common tile size is the GCF. Toggle to LCM mode to see two skip-count number lines: the first tick marked on both lines is the LCM.
🔑 Prime factorisation method (for larger numbers)

GCF: prime-factorise both numbers; multiply the shared prime factors (using the lower exponent for each).

Example: 36 = 2² × 3², 48 = 2⁴ × 3. GCF = 2² × 3 = 12.

LCM: multiply each prime factor at its higher exponent.

Same example: LCM = 2⁴ × 3² = 16 × 9 = 144.

Check: GCF × LCM = 12 × 144 = 1,728 = 36 × 48 ✓

✨ GCF × LCM = a × b — always

For any two positive integers a and b: GCF(a, b) × LCM(a, b) = a × b. This means if you know one, you can calculate the other without listing factors again.

Example: GCF(8, 12) = 4. So LCM = (8 × 12) ÷ 4 = 96 ÷ 4 = 24.

📝 Worked example: Find GCF(24, 36) and LCM(24, 36).
  1. Prime factorise: 24 = 2³ × 3, 36 = 2² × 3².
  2. GCF: take lower exponent for each prime → 2² × 3¹ = 4 × 3 = 12.
  3. LCM: take higher exponent for each prime → 2³ × 3² = 8 × 9 = 72.
  4. Check: 12 × 72 = 864 = 24 × 36 ✓
✓ GCF = <strong>12</strong>, LCM = <strong>72</strong>.

Check your understanding

1. What is GCF(20, 30)?
Factors of 20: 1,2,4,5,10,20. Factors of 30: 1,2,3,5,6,10,15,30. Largest common: 10.
2. What is LCM(6, 8)?
Multiples of 6: 6,12,18,24... Multiples of 8: 8,16,24... First common: 24.
3. If GCF(a, b) = 6 and a × b = 216, what is LCM(a, b)?
LCM = a × b ÷ GCF = 216 ÷ 6 = 36.
4. You have 16 red marbles and 24 blue marbles. You want to put them into equal bags with no leftover. What is the largest number of bags you can make?
GCF(16, 24) = 8. The largest number of equal bags is 8 (2 red + 3 blue each).
5. Bus A comes every 8 minutes, bus B every 10 minutes. If both arrive at 9:00 am, when is the next time they arrive together?
LCM(8, 10) = 40 minutes. So both buses arrive together again at 9:00 + 40 min = 9:40 am. Wait — that's option A: 9:40 am. LCM(8,10): multiples of 8: 8,16,24,32,40; multiples of 10: 10,20,30,40. LCM = 40. 9:00 + 40 min = 9:40 am.
✅ Key takeaways
  • GCF is the largest number that divides both a and b evenly — useful for splitting into equal groups.
  • LCM is the smallest positive number that both a and b divide into — useful for finding when cycles sync.
  • Prime factorisation: GCF uses the lower exponent; LCM uses the higher exponent for each prime.
  • GCF(a, b) × LCM(a, b) = a × b — use this shortcut when one value is known.
  • GCF ≤ min(a, b); LCM ≥ max(a, b) — sanity-check your answers.