HCF & LCM

Highest Common Factor (HCF)

A common factor of two numbers is a number that both numbers can be divided by
- 1 is a common factor of any two numbers
To find common factors you can write out the factors of each number and identify the numbers that appear in both lists
- The factors of a common factor of two numbers will also be common factors
  - 6 is a common factor of 24 and 30
  - Therefore 1, 2, 3, 6 are also common factors of 24 and 30
The highest common factor is the largest common factor between two numbers
- This is useful when simplifying fractions or factorising expressions

Write each number as a product of its prime factors
Find the prime factors that are common to both numbers
- Be careful with how many times a prime factor appears
  - 12 = 2 × 2 × 3 and 10 = 2 × 5
  - So only one of the 2s is a common prime factor
Multiply the common prime factors together
Using a Venn diagram can help
- Put the common prime factors in the centre
- Put the other prime factors in the relevant circles
- The HCF is the product of all the numbers in the centre

Find the highest common factor of 42 and 90.

Write as a product of prime factors

42 = 2 × 3 × 7
90 = 2 × 3 × 3 × 5

Write the prime factors in a Venn diagram if needed.

Venn-Diagram-42-90, IGCSE & GCSE Maths revision notes

Multiply the common prime factors.

HCF = 2 × 3

HCF = 6

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
To find common multiples you can write out the multiples of each number and identify the numbers that appear in both lists
- The multiples of a common multiple of two numbers will also be common multiples
  - 60 is a common multiple of 12 and 10
  - Therefore 60, 120, 180, 240, etc are also common multiples of 12 and 10
The lowest common multiple is the smallest common multiple between two numbers
- This is useful when adding or subtracting fractions

Write each number as a product of its prime factors
Find the prime factors of the first number that are not prime factors of the second number
- Be careful with how many times a prime factor appears
  - 12 = 2 × 2 × 3 and 10 = 2 × 5
  - So 3 and one of the 2s are not prime factors of the 10
  - Equivalently 5 is not a prime factor of 12
Multiply the first number by these extra prime factors
- Either multiply 10 by 3 and 2
- Or multiply 12 by 5
- Both ways will give you the same answer
Using a Venn diagram can help
- Put the common prime factors in the centre
- Put the other prime factors in the relevant circles
- The LCM is the product of all the numbers in the Venn diagram