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§ Arithmetic

Factors, GCF & LCM

§ Arithmetic

Factors, GCF & LCM

CCSS.6.NS3 min read

Factors are whole numbers that divide evenly into another number, whilst multiples are the results of multiplying a number by whole numbers. The Greatest Common Factor (GCF) identifies the largest number that divides two or more numbers, and the Lowest Common Multiple (LCM) finds the smallest number that both original numbers divide into evenly.

§ 01

Why it matters

GCF and LCM calculations appear throughout mathematics and practical situations. When simplifying fractions like 2436, finding GCF(24,36) = 12 reduces it to 23. Recipe scaling relies on LCM — if one recipe serves 6 people and another serves 8, LCM(6,8) = 24 tells us the smallest group size for equal portions. Timetabling uses LCM to find when events coincide: buses arriving every 15 minutes and trains every 20 minutes both arrive together every LCM(15,20) = 60 minutes. GCF helps distribute items equally — sharing 48 sweets among 18 children requires GCF(48,18) = 6 groups of equal size. These concepts underpin algebraic fraction work in GCSE mathematics and appear in Year 5 and Year 7 curriculum requirements.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

What is the GCF of 8 and 9?

Answer: 1

  1. List factors of 8 → [1, 2, 4, 8] — Find all numbers that divide evenly.
  2. List factors of 9 → [1, 3, 9] — Same for the second number.
  3. Find greatest common → GCF = 1 — The largest number in both lists.
Easy§ 02

What is the GCF of 24 and 36?

Answer: 12

  1. Use prime factorisation → GCF(24, 36) — Factor both numbers into primes.
  2. Find common prime factors → GCF = 12 — Multiply the shared primes.
  3. Verify → 24 ÷ 12 = 2, 36 ÷ 12 = 3 ✓ — Both divide evenly by the GCF.
Medium§ 03

What is the GCF of 45 and 48?

Answer: 3

  1. Use prime factorisation → GCF(45, 48) — Factor both numbers into primes.
  2. Find common prime factors → GCF = 3 — Multiply the shared primes.
  3. Verify → 45 ÷ 3 = 15, 48 ÷ 3 = 16 ✓ — Both divide evenly by the GCF.
§ 04

Common mistakes

  • Confusing GCF with LCM, such as stating GCF(12,18) = 36 instead of 6, or claiming LCM(12,18) = 6 instead of 36
  • Missing factors when listing systematically, like writing factors of 24 as {1,2,3,4,6,8,12} and omitting 24 itself
  • Calculating LCM by multiplying the original numbers without dividing by GCF, giving LCM(8,12) = 96 instead of 24
§ 05

Frequently asked questions

What is the difference between GCF and LCM?
GCF finds the largest number that divides into both given numbers, whilst LCM finds the smallest number that both given numbers divide into. For numbers 12 and 18: GCF = 6 (largest shared factor), LCM = 36 (smallest shared multiple).
How do you find GCF using prime factorisation?
Write both numbers as products of prime factors, then multiply the common prime factors together. For GCF(24,36): 24 = 2³×3, 36 = 2²×3². Common factors are 2² and 3¹, so GCF = 4×3 = 12.
What is the GCF of two prime numbers?
The GCF of two different prime numbers is always 1, since prime numbers have no factors other than 1 and themselves. For example, GCF(7,11) = 1 because neither number shares any factors with the other.
How do you calculate LCM without listing multiples?
Use the formula LCM(a,b) = (a×b) ÷ GCF(a,b). For LCM(15,20): first find GCF(15,20) = 5, then calculate LCM = (15×20) ÷ 5 = 300 ÷ 5 = 60.
Can the GCF be larger than one of the original numbers?
No, the GCF cannot exceed the smaller of the two numbers. The GCF is always less than or equal to the smallest number being compared, since factors divide evenly into the original number.
§ 06

See also

FactorMultipleFraction
§ 06

Where to next?

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