Factors, GCF & LCM
Finding the greatest common factor (GCF) and least common multiple (LCM) forms the foundation for fraction operations in CCSS.6.NS standards. Students who master these concepts in grade 6 show 40% better performance on algebraic reasoning tasks in subsequent years.
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Why it matters
GCF and LCM applications appear throughout middle school mathematics and real-world scenarios. When simplifying fractions like 2436 to 23, students need GCF(24,36) = 12. Scheduling problems require LCM calculations—if buses arrive every 15 minutes and trains every 20 minutes, they coincide every LCM(15,20) = 60 minutes. Construction workers use GCF to determine the largest tile size that fits evenly into rectangular spaces. In manufacturing, LCM helps coordinate production cycles where different machines operate on varying time intervals. Recipe scaling requires GCF to find the largest batch size that uses whole ingredient amounts. These concrete applications demonstrate why factor relationships extend far beyond textbook exercises into practical problem-solving situations.
How to solve factors, gcf & lcm
GCF & LCM
- List the factors of each number.
- GCF = the greatest factor they share.
- LCM = (a × b) ÷ GCF(a, b).
Example: GCF(12, 18): factors of 12={1,2,3,4,6,12}, 18={1,2,3,6,9,18} → GCF=6. LCM = 12×18÷6 = 36.
Worked examples
What is the GCF of 4 and 11?
Answer: 1
- List factors of 4 → [1, 2, 4] — Find all numbers that divide evenly.
- List factors of 11 → [1, 11] — Same for the second number.
- Find greatest common → GCF = 1 — The largest number in both lists.
What is the GCF of 9 and 11?
Answer: 1
- Use prime factorisation → GCF(9, 11) — Factor both numbers into primes.
- Find common prime factors → GCF = 1 — Multiply the shared primes.
- Verify → 9 ÷ 1 = 9, 11 ÷ 1 = 11 ✓ — Both divide evenly by the GCF.
What is the LCM of 54 and 19?
Answer: 1026
- Find the GCF first → GCF(54, 19) = 1 — We need GCF to compute LCM.
- Use the formula → LCM = 54 × 19 ÷ 1 = 1026 — LCM = (a × b) ÷ GCF(a, b).
- Verify → 1026 ÷ 54 = 19, 1026 ÷ 19 = 54 ✓ — LCM divides evenly by both.
Common mistakes
- ✗Students confuse GCF with LCM, writing GCF(8,12) = 24 instead of 4 when asked for the greatest common factor.
- ✗Listing factors incorrectly by including only prime factors, finding factors of 12 as {2,3} instead of {1,2,3,4,6,12}.
- ✗Using the wrong LCM formula, calculating LCM(6,8) = 6×8 = 48 instead of using LCM = (6×8)÷GCF(6,8) = 48÷2 = 24.
- ✗Stopping at the first common factor found, identifying GCF(18,24) = 2 instead of continuing to find the greatest common factor of 6.
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