Factors, GCF & LCM
Factors are whole numbers that divide evenly into another number without leaving a remainder. The Greatest Common Factor (GCF) represents the largest number that divides into two or more numbers, while the Least Common Multiple (LCM) is the smallest number that both original numbers divide into evenly. These concepts appear in CCSS Grade 4 standards for finding factor pairs and identifying prime and composite numbers.
Why it matters
Factors, GCF, and LCM solve practical problems across multiple domains. In construction, finding the GCF of measurements like 24 inches and 36 inches (GCF = 12) helps determine the largest tile size that fits evenly. Scheduling problems use LCM — if one bus arrives every 15 minutes and another every 20 minutes, they meet every 60 minutes (LCM of 15 and 20). These concepts underpin fraction operations, since adding 112 + 118 requires finding LCM(12,18) = 36 for the common denominator. In algebra, factoring polynomials builds directly on number factoring skills. Manufacturing uses GCF to determine efficient packaging sizes, while computer science applies these concepts in algorithm optimization and data structure design.
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 6 and 4?
Answer: 2
- List factors of 6 → [1, 2, 3, 6] — Find all numbers that divide evenly.
- List factors of 4 → [1, 2, 4] — Same for the second number.
- Find greatest common → GCF = 2 — The largest number in both lists.
What is the GCF of 13 and 12?
Answer: 1
- Use prime factorisation → GCF(13, 12) — Factor both numbers into primes.
- Find common prime factors → GCF = 1 — Multiply the shared primes.
- Verify → 13 ÷ 1 = 13, 12 ÷ 1 = 12 ✓ — Both divide evenly by the GCF.
What is the GCF of 41 and 9?
Answer: 1
- Use prime factorisation → GCF(41, 9) — Factor both numbers into primes.
- Find common prime factors → GCF = 1 — Multiply the shared primes.
- Verify → 41 ÷ 1 = 41, 9 ÷ 1 = 9 ✓ — Both divide evenly by the GCF.
Common mistakes
- Confusing GCF and LCM leads to writing GCF(8,12) = 24 instead of 4, mixing up the greatest common factor with the least common multiple.
- Missing factors when listing creates incomplete factor sets, such as listing factors of 12 as {1,2,3,6,12} while omitting 4.
- Using the LCM formula incorrectly results in calculating LCM = (a × b) × GCF instead of LCM = (a × b) ÷ GCF, producing 12 × 18 × 6 = 1296 rather than 36.