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Number · Foundation & Higher

Factors, multiples & primes

Factors divide exactly into a number; multiples are in a number's times table. Prime numbers have exactly two factors: 1 and themselves. HCF and LCM are key skills for GCSE.

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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

HCF from prime factors
Multiply shared prime factors (lowest powers)

e.g. HCF(12, 18) = 2 × 3 = 6

LCM from prime factors
Multiply all prime factors (highest powers)

e.g. LCM(12, 18) = 2² × 3² = 36

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Worked examples

Example 1

Find the HCF and LCM of 36 and 60.

Working

  1. 36 = 2² × 3²
  2. 60 = 2² × 3 × 5
  3. HCF: shared factors with lowest powers = 2² × 3 = 12
  4. LCM: all factors with highest powers = 2² × 3² × 5 = 180
AnswerHCF = 12, LCM = 180
Example 2

Express 420 as a product of its prime factors.

Working

  1. 420 ÷ 2 = 210
  2. 210 ÷ 2 = 105
  3. 105 ÷ 3 = 35
  4. 35 ÷ 5 = 7
  5. 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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