HCF & LCM

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

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 

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

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

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

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

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 circles

• The LCM of 42 and 90 is 7×2×3×3×5, which is 630

Write each number as a product of the powers of its prime factors

• $72=2^3\times 3^2$ and $540 = 2^2\times 3^3\times 5$

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