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§ Arithmetic

Factors, GCF & LCM

CCSS.6.NS3 min read

Year 7 students frequently struggle with prime factorisation when finding HCF and LCM, often confusing the two concepts entirely. The National Curriculum requires mastery of these skills by Year 7, building on Year 5 foundations of identifying multiples and factors.

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§ 01

Why it matters

Factors, GCF, and LCM appear throughout GCSE maths and real-world scenarios. Students need GCF to simplify fractions like 1824 = 34, whilst LCM helps calculate when events coincide—if Harry's bus arrives every 15 minutes and Isla's every 20 minutes, they'll meet again after 60 minutes (LCM of 15 and 20). In practical contexts, GCF determines the largest equal groups possible: dividing 48 chocolates and 36 sweets equally gives 12 portions of 4 chocolates and 3 sweets each. Year 7 pupils use prime factorisation to tackle larger numbers systematically, essential for algebraic manipulation and ratio problems in higher years. These concepts underpin fraction operations, problem-solving with time intervals, and geometric applications involving tessellations.

§ 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 5 and 9?

Answer: 1

  1. List factors of 5 → [1, 5] — 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 11?

Answer: 1

  1. Use prime factorisation → GCF(24, 11) — Factor both numbers into primes.
  2. Find common prime factors → GCF = 1 — Multiply the shared primes.
  3. Verify → 24 ÷ 1 = 24, 11 ÷ 1 = 11 ✓ — Both divide evenly by the GCF.
Medium§ 03

What is the GCF of 37 and 57?

Answer: 1

  1. Use prime factorisation → GCF(37, 57) — Factor both numbers into primes.
  2. Find common prime factors → GCF = 1 — Multiply the shared primes.
  3. Verify → 37 ÷ 1 = 37, 57 ÷ 1 = 57 ✓ — Both divide evenly by the GCF.
§ 04

Common mistakes

  • Students confuse GCF with LCM, writing GCF(12, 8) = 24 instead of 4, mixing up 'greatest common' with 'least common'
  • Pupils list multiples instead of factors, finding 'factors of 12' as 12, 24, 36 rather than 1, 2, 3, 4, 6, 12
  • When using prime factorisation, students multiply all prime factors instead of only common ones, calculating GCF(18, 24) = 2×3×3×2×2×2 = 144 instead of 2×3 = 6
  • Students forget the LCM formula, attempting LCM(8, 12) = 8 + 12 = 20 instead of using (8×12) ÷ GCF(8,12) = 96 ÷ 4 = 24
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§ 05

Frequently asked questions

What's the difference between factors and multiples?
Factors divide into a number evenly (factors of 12: 1, 2, 3, 4, 6, 12), whilst multiples are found by multiplying (multiples of 12: 12, 24, 36, 48). Think factors go 'into' numbers, multiples go 'beyond' them.
When should Year 7s use prime factorisation versus listing?
Prime factorisation works efficiently for larger numbers (over 30). For GCF(48, 72), listing factors becomes unwieldy, but 48 = 2⁴×3 and 72 = 2³×3² gives GCF = 2³×3 = 24 quickly.
How do I explain LCM using the GCF formula?
LCM(a,b) = (a×b) ÷ GCF(a,b) works because you're removing the 'double-counted' common factors. For LCM(15,20): 15×20 = 300, but GCF = 5, so LCM = 300÷5 = 60.
What if the GCF is 1?
When GCF = 1, numbers are coprime (share no common factors except 1). This happens with consecutive numbers like 7 and 8, or when one number is prime like 5 and 9. LCM then equals their product.
How does this link to fractions in Year 8?
GCF simplifies fractions to lowest terms: 18/24 ÷ GCF(18,24) = 18/24 ÷ 6 = 3/4. LCM finds common denominators for adding unlike fractions: 1/6 + 1/4 needs denominator LCM(6,4) = 12.
§ 06

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