IGCSEMaths: CoreCIERevision NotesNumberPrime Factors, HCF & LCMHCF & LCM

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HCF & LCM (CIE IGCSE Maths: Core)

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CIE Maths: CoreEdexcel Maths A (Modular): Higher Unit 2Cambridge (CIE) International Maths: ExtendedCambridge (CIE) International Maths: Core

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 always 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 

How can I use a Venn diagram to find the highest common factor (HCF) of two numbers?

  • 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 2 in the centre of the diagram
    • Although 3 appears twice in the prime factors of 90, it appears once in the prime factors of 42
      • Put a single 3 in the centre of the diagram
    • If there are no common prime factors, put a 1 in the centre of the diagram
  • 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
  • If there are no common prime factors then the HCF is 1

Venn diagram of prime factors for 42 and 90

How can I use the powers of prime factors to find the highest common factor (HCF) of two numbers?

  • Write each number as a product of the powers of its prime factors
    • 24 = 23×3 and 60 = 22×3×5
  • Find all common prime factors and identify the highest power that appears in both numbers
    • The highest power of 2 in both is 22
      • 22 is a common factor
    • The highest power of 3 in both is 31
      • 3 is a common factor
    • No other prime number appears in both
  • The highest common factor is the product of the common powers of primes
      • The HCF of 24 and 60 is 22×3 which is 12

Examiner 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

Another alternative is to find the highest common powers of primes

22 and 31 are the highest common powers of primes
HCF = 22 × 31

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 original 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 multiples 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 until you find a common multiple
  • The lowest common multiple will be the smallest multiple that appears in both lists

How can I use a Venn diagram to find the lowest common multiple (LCM) of two numbers?

  • 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 a 2 in the centre of the diagram
    • Although 3 appears twice in the prime factors of 90, it appears once in the prime factors of 42
      • Put a single 3 in the centre of the diagram
    • If there are no common prime factors then put a 1 in the centre of the diagram
  • 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

How can I use the powers of prime factors to find the lowest common multiple (LCM) of two numbers?

  • Write each number as a product of the powers of its prime factors
    • 72 = 23×32 and 540 = 22×33×5
  • Find the highest power of each and every prime that appears in both numbers (they do not have to be common primes)
    • 23 is the highest power of 2 shown (from 72)
    • 33 is the highest power of 3 shown (from 540)
    • 51 is the highest power of 5 shown (from 540)
      • Note: 5 is not a common prime in 72 and 540, but that does not matter for LCMs
  • The lowest common multiple is the product of these highest powers 
    • The LCM of 72 and 540 is 23×33×5, which is 1080

Examiner 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

An alternative method is to write out the multiples

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

Another alternative method is to find the highest powers of each and every prime that appear
Then multiply these together

23 × 32 × 51

LCM = 360

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    Jamie W

    Author: Jamie W

    Expertise: Maths

    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.