Revising factors, multiples & primes the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Factors, multiples & primes GCSE” on the site for the complete guide.
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Key facts to remember
- 1A factor of n divides into n with no remainder.
- 2A multiple of n is in n's times table (n, 2n, 3n, …).
- 3A prime number has exactly two factors: 1 and itself. 1 is NOT prime.
- 4Prime factorisation: write a number as a product of its prime factors using a factor tree.
- 5HCF (Highest Common Factor): the largest factor shared by two numbers.
- 6LCM (Lowest Common Multiple): the smallest multiple shared by two numbers.
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Formulas
Multiply shared prime factors (lowest powers)e.g. HCF(12, 18) = 2 × 3 = 6
Multiply all prime factors (highest powers)e.g. LCM(12, 18) = 2² × 3² = 36
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Worked examples
Example 1Find the HCF and LCM of 36 and 60.
Working
- 36 = 2² × 3²
- 60 = 2² × 3 × 5
- HCF: shared factors with lowest powers = 2² × 3 = 12
- LCM: all factors with highest powers = 2² × 3² × 5 = 180
AnswerHCF = 12, LCM = 180
Example 2Express 420 as a product of its prime factors.
Working
- 420 ÷ 2 = 210
- 210 ÷ 2 = 105
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 420 = 2² × 3 × 5 × 7
Answer2² × 3 × 5 × 7
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Common mistakes
✗Including 1 as a prime number — 1 is not prime because it only has one factor.
✗Confusing HCF and LCM: HCF is always ≤ the smaller number; LCM is always ≥ the larger number.
✗Incomplete factor trees — not breaking numbers down fully to prime factors.
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Exam tips
✓Use a Venn diagram to find HCF and LCM: shared primes go in the middle (multiply for HCF), all primes multiplied give LCM.
✓Check prime factorisation by multiplying back: your product should equal the original number.
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