GCSEMaths: FoundationEdexcelRevision NotesNumberTypes of Number, Prime Factors, HCF & LCMHCF & LCM

HCF & LCM (Edexcel GCSE Maths: Foundation)

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Jamie W

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Jamie W

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Highest Common Factor (HCF)

What is the highest common factor (HCF) of two numbers?

  • A common factor of two numbers is a value that both numbers can be divided by, leaving no remainder
    • 1 is a common factor of any two numbers
    • Any factor of a common factor will also be a common factor of the original two numbers
      • 6 is a common factor of 24 and 30
      • Therefore 1, 2 and 3 are also common factors of 24 and 30
  • The highest common factor is the largest common factor of the two numbers
    • The highest common factor is useful when simplifying fractions or factorising expressions

How do I find the highest common factor (HCF) of two numbers?

  • To find common factors:
    • write out the factors of each number in a list
    • identify the numbers that appear in both lists
  • The highest common factor will be the largest factor that appears in both lists 
  • A Venn diagram method can be used instead of listing out all the factors
    • Write each number as a product of its prime factors
      • 42 = 2×3×7 and 90 = 2×3×3×5
    • Find the prime factors that are common to both numbers and put these in the centre of the Venn diagram
      • 42 and 90 both have a prime factor of 2; put this in the centre
      • Although 3 appears twice in the prime factors of 90, 3 only appears once in the prime factors of 42
      • So only put a single 3 in the centre of the diagram
      • If there are no common prime factors then put a 1 in the centre
    • Put the remaining prime factors in the respective regions
      • 7 would go in the region for 42
      • 3 and 5 would go in the region for 90
    • The highest common factor is the product of the numbers in the centre
      • The HCF of 42 and 90 is 2×3, which is 6

Venn diagram of prime factors for 42 and 90

Exam Tip

  • The highest common factor of two numbers could be one of the numbers!
    • The highest common factor of 4 and 12 is 4

Worked example

Find the highest common factor of 36 and 120.

Write both numbers as a product of prime factors

36 = 2×2×3×3 = 22 × 32
120 = 2×2×2×3×5 = 23 × 3 × 5

Write the prime factors in a Venn diagram

Venn diagram of prime factors of 36 and 120

Multiply the common prime factors in the centre

HCF = 2 × 2 × 3

Alternatively list the factors for each number

36: 1, 2, 3, 4, 6, 9, 12, 18, 36
120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

HCF = 12

Lowest Common Multiple (LCM)

What is the lowest common multiple (LCM) of two numbers?

  • A common multiple of two numbers is a number that appears in both of their times tables
    • The product of the two numbers is always a common multiple (but not necessarily the lowest)
    • Any multiple of a common multiple will also be a common multiple of the original two numbers
      • 30 is a common multiple of 3 and 10
      • Therefore 60, 90, 120, ... are also common factors of 3 and 10
  • The lowest common multiple is the smallest common multiple between two numbers
    • This is useful when finding a common denominator and when adding or subtracting fractions

How do I find the lowest common multiple (LCM) of two numbers?

  • To find the lowest common multiple of two numbers:
    • write out the first few multiples of each number
    • identify the multiples that appear in both lists
      • If there are none then write out the next few multiples of each number
  • The lowest common multiple will be the smallest multiple that appears in both lists
  • A Venn diagram method can be used instead of writing out multiples
    • Write each number as a product of its prime factors
      • 42 = 2×3×7 and 90 = 2×3×3×5
    • Find the prime factors that are common to both numbers and put these in the centre of the Venn diagram
      • 42 and 90 both have a prime factor of 2; put this in the centre
      • Although 3 appears twice in the prime factors of 90, 3 only appears once in the prime factors of 42
      • So only put a single 3 in the centre of the diagram
      • If there are no common prime factors then put a 1 in the centre
    • Put the remaining prime factors in the respective regions
      • 7 would go in the region for 42
      • 3 and 5 would go in the region for 90
    • The lowest common multiple is the product of all the numbers in the Venn diagram
      • The LCM of 42 and 90 is 7×2×3×3×5, which is 630

Venn diagram of prime factors for 42 and 90

Exam Tip

  • The lowest common multiple of two numbers could be one of the numbers!
    • The lowest common multiple of 4 and 12 is 12

Worked example

Find the lowest common multiple of 36 and 120.

Write both numbers as a product of prime factors

36 = 2×2×3×3 = 22 × 32
120 = 2×2×2×3×5 = 23 × 3 × 5

Write the prime factors in a Venn diagram

Venn diagram of prime factors of 36 and 120

Multiply all the prime factors in the diagram

LCM = 3 × 2 × 2 × 3 × 2 × 5

Alternatively write out the multiples

36: 26, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, ...
120: 120, 240, 360, 480, ...

LCM = 360

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    Jamie W

    Author: Jamie W

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.