HCF & LCM (Edexcel IGCSE Maths)

Highest Common Factor (HCF)

Spec point — 1.4E

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

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

Exam Hint The highest common factor of two numbers could be one of the numbers!

E.g. The highest common factor of 4 and 12 is 4.

Worked Example Find the highest common factor of 36 and 120.

Answer:

Write both numbers as a product of prime factors

36 = 2×2×3×3 = 2 ² × 3 ² 120 = 2×2×2×3×5 = 2³**× 3 × 5

Write the prime factors in a Venn diagram

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

2*²and 3¹are the highest common powers of primes HCF = 2²× 3¹*

Final answer: HCF = 12

Lowest Common Multiple (LCM)

Spec point — 1.4E

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 circles

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

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

  • $72=2^{3}\times3^{2}$ and $540=2^{2}\times3^{3}\times5$

• 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

Exam Hint The lowest common multiple of two numbers could be one of the numbers!

E.g. The lowest common multiple of 4 and 12 is 12.

Worked Example Find the lowest common multiple of 36 and 120.

Answer:

Write both numbers as a product of prime factors

36 = 2×2×3×3 = 2 ² × 3 ² 120 = 2×2×2×3×5 = 2³**× 3 × 5

Write the prime factors in a Venn diagram

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: 36, 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 appears Then multiply these together

2*³× 3²× 5¹*

Final answer: LCM = 360