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Mathematics 05801.1

Types of number, HCF and LCM

Primes, factors, multiples, and finding HCF and LCM by prime factorisation.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 1.1.1Identify and use natural numbers, primes, factors, multiples, squares and cubes.
  • 1.1.2Express numbers as products of prime factors and find the HCF and LCM.

5 minute read

Types of number, HCF and LCM

The vocabulary

  • Factor — a number that divides exactly into another. Factors of 12: 1, 2, 3, 4, 6, 12.
  • Multiple — the result of multiplying by an integer. Multiples of 12: 12, 24, 36, …
  • Prime — exactly two factors, itself and 1. So 2, 3, 5, 7, 11, … Note 1 is not prime (it has only one factor) and 2 is the only even prime.
  • Square numbers 1, 4, 9, 16; cube numbers 1, 8, 27, 64.

Prime factorisation

Every integer above 1 breaks into a unique product of primes. Use a factor tree, dividing by the smallest prime that works each time: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

HCF and LCM from prime factors

Write both numbers in prime factor form, then:

  • HCF (highest common factor) — multiply the common primes, taking the lower power of each.
  • LCM (lowest common multiple) — multiply all primes appearing, taking the higher power of each.

For 60 = 2²×3×5 and 72 = 2³×3²:

  • HCF = 2² × 3 = 12
  • LCM = 2³ × 3² × 5 = 360

A useful check: HCF × LCM = product of the two numbers. Here 12 × 360 = 4320 = 60 × 72 ✓

Which one does a word problem want?

  • Things happening together again (buses, bells, laps) → LCM.
  • Splitting into equal largest groups with nothing left over → HCF.

Think of it like this

Prime factors are a number's DNA. Once both numbers are written in that form, the HCF is what they share and the LCM is everything either of them needs — no guessing required.

Worked examples

Method, step by step

Find the HCF and LCM of 24 and 36.

  1. 1Prime factorise: 24 = 2³ × 3 and 36 = 2² × 3².
  2. 2HCF: take common primes to the lower power — 2² × 3 = 12.
  3. 3LCM: take all primes to the higher power — 2³ × 3² = 72.
  4. 4Check: 12 × 72 = 864 = 24 × 36 ✓

HCF = 12, LCM = 72

Common misconceptions

  • Calling 1 a prime number. A prime has exactly two distinct factors; 1 has one.
  • Taking the higher power for HCF. HCF uses the **lower** power of each shared prime.
  • Listing multiples by hand to find the LCM of large numbers, which is slow and error-prone compared with prime factorisation.

In the exam

  • Use the HCF × LCM = product check whenever both are asked for — it catches an error in seconds.
  • Read word problems for the signal: "at the same time again" means LCM, "largest equal groups" means HCF.