Year 9: Highest common factor and lowest common multiple

What are the Highest Common Factor (HCF) and Lowest Common Multiple (LCM)?

The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) are fundamental concepts in Number Theory. They are closely related and often used together to solve problems involving sets of numbers.

Highest Common Factor (HCF)

Definition

The HCF of two or more numbers is the largest number that divides evenly into all of them.

Finding the HCF

• Prime Factorisation: Break each number into its prime factors. The HCF is the product of the common prime factors raised to the lowest power.
• Listing Factors: List all the factors of each number and identify the largest one that appears in all lists.

Example: HCF of 12 and 18

12 = 2 x 2 x 3 and 18 = 2 x 3. Therefore, HCF(12, 18) = 2 x 3 = 6

Lowest Common Multiple (LCM)

Definition

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Finding the LCM

• Prime Factorisation: Break each number into its prime factors. The LCM is the product of all prime factors raised to the highest power.
• Using HCF: LCM(a, b) = (a x b) / HCF(a, b)

Example: LCM of 12 and 18

12 = 2 x 2 x 3 and 18 = 2 x 3. Therefore, LCM(12, 18) = 2 x 3 = 6