HCF & LCM | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Highest Common Factor (HCF)

What is the highest common factor (HCF) of two numbers?

• A common factor of two numbers is a number that both numbers can be divided by. For example, 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. For instance, 6 is a common factor of 24 and 30. Therefore, 1, 2, 3, and 6 are also common factors of 24 and 30.
• The highest common factor is the largest common factor between two numbers. This is particularly useful when simplifying fractions or factorizing expressions.

How do I find the highest common factor (HCF) of two numbers?

• To find the highest common factor (HCF) of two numbers:
• Express each number as a product of its prime factors.
  • Identify the prime factors that are common to both numbers, ensuring to account for the number of times a prime factor appears.
  • For instance, consider 12 (2 × 2 × 3) and 10 (2 × 5). Only one '2' is a common prime factor.
• Multiply the common prime factors together. 
• Using a Venn diagram can assist in visualizing this process.
  • Place the common prime factors in the center of the diagram and the remaining prime factors in the relevant circles.
  • The HCF is determined as the product of all the numbers in the center of the Venn diagram.

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 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

How do I find the lowest common multiple (LCM) of two numbers?

• Write each number as a product of its prime factors
  • Breaking down numbers into their prime factors is a crucial step in understanding their unique makeup. For instance, consider the number 12. Its prime factors are 2 x 2 x 3. Similarly, the prime factors of 10 are 2 x 5.
• Find the prime factors of the first number that are not prime factors of the second number
  • Identifying the prime factors that are exclusive to one number can be insightful. For example, when comparing 12 and 10, we notice that 3 and one of the 2s are unique to 12, while 5 is specific to 10.
• Be cautious with the occurrences of prime factors
  • Understanding how many times a prime factor appears is crucial. In the case of 12 (2 x 2 x 3) and 10 (2 x 5), we see that 3 and one of the 2s are not shared between the two numbers. Similarly, 5 is not a prime factor of 12.
• Multiply the first number by these extra prime factors
  • When extra prime factors are identified, such as 3 and 2 for 10 or 5 for 12, multiplying the first number by these additional factors yields the least common multiple (LCM). For example, you can multiply 10 by 3 and 2 or 12 by 5 to achieve the same LCM.
• Using visual aids like Venn diagrams for better comprehension
  • Venn diagrams can effectively illustrate the common and unique prime factors between two numbers. Placing common factors in the intersection and unique factors in the respective circles aids in understanding. The LCM is then derived from the product of all elements in the Venn diagram.