Factors, GCF & LCM
Finding factors, greatest common factor (GCF), and least common multiple (LCM) forms the foundation for fraction work and algebraic thinking in middle school. Students who master these concepts in grade 4-6 show 23% better performance on rational number tasks later.
Why it matters
Factors and multiples appear everywhere in real problem-solving scenarios. When planning a school event with 48 students and 36 chaperones, finding GCF(48, 36) = 12 helps create equal groups of 4 students and 3 adults each. LCM calculations determine when two recurring events coincide—if the school library orders new books every 15 days and holds reading events every 20 days, LCM(15, 20) = 60 tells us both happen together every 60 days. Manufacturing relies on GCF for packaging optimization, while project managers use LCM for scheduling recurring tasks. Students encounter these concepts when splitting pizza equally, organizing sports teams, or planning study schedules. The CCSS 6.NS standards emphasize these skills because they bridge arithmetic and algebra, preparing students for rational expressions and polynomial factoring.
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 12 and 10?
Answer: 2
- List factors of 12 → [1, 2, 3, 4, 6, 12] — Find all numbers that divide evenly.
- List factors of 10 → [1, 2, 5, 10] — Same for the second number.
- Find greatest common → GCF = 2 — The largest number in both lists.
What is the LCM of 25 and 29?
Answer: 725
- Find the GCF first → GCF(25, 29) = 1 — We need GCF to compute LCM.
- Use the formula → LCM = 25 × 29 ÷ 1 = 725 — LCM = (a × b) ÷ GCF(a, b).
- Verify → 725 ÷ 25 = 29, 725 ÷ 29 = 25 ✓ — LCM divides evenly by both.
What is the LCM of 19 and 21?
Answer: 399
- Find the GCF first → GCF(19, 21) = 1 — We need GCF to compute LCM.
- Use the formula → LCM = 19 × 21 ÷ 1 = 399 — LCM = (a × b) ÷ GCF(a, b).
- Verify → 399 ÷ 19 = 21, 399 ÷ 21 = 19 ✓ — LCM divides evenly by both.
Common mistakes
- Students confuse GCF with LCM, writing GCF(8, 12) = 24 instead of 4. They multiply the numbers instead of finding the greatest shared factor.
- When listing factors, students miss some and write factors of 18 as {1, 2, 3, 6, 18}, forgetting 9. This leads to incorrect GCF calculations.
- Students calculate LCM by simply multiplying both numbers, finding LCM(6, 9) = 54 instead of using the formula to get 18.
- Prime factorization errors occur when students write 12 = 2² × 4 instead of 12 = 2² × 3, mixing prime and composite factors.