HCF & LCM (Cambridge (CIE) IGCSE Maths) : Revision Note

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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 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 Tips and Tricks

  • 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 equals 2 cubed cross times 3 squared and 540 space equals space 2 squared cross times 3 cubed cross times 5

  • Find the highest power of each and every prime that appears in either number (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 it is still needed for the LCM

  • 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 Tips and Tricks

  • 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 Wood

Author: Jamie Wood

Expertise: Maths Content Creator

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.

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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