Highest Common Factor (HCF)
The highest common factor (HCF) of two numbers is the largest factor that divides both
numbers.
● A common factor is any number that can divide both numbers without leaving a
remainder.
● The number 1 is always a common factor of any two numbers.
● To identify common factors, list all factors of each number and find the ones that
appear in both lists.
For example:
● The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
● The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
● The common factors are 1, 2, 3, and 6.
● The highest common factor is 6.
Finding the HCF Using Prime Factorization
1. Write each number as a product of its prime factors.
2. Identify the prime factors that are common to both numbers.
3. Multiply the common prime factors to find the HCF.
Example:
Find the HCF of 42 and 90.
● Prime factorization:
○ 42 = 2 × 3 × 7
○ 90 = 2 × 3 × 3 × 5
● Common prime factors: 2 and 3
● HCF = 2 × 3 = 6
A Venn diagram can help by placing the common prime factors in the center and the unique
factors in separate circles. The product of the center values gives the HCF.
Lowest Common Multiple (LCM)
The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple
of both.
● A common multiple appears in the times tables of both numbers.
● The product of the two numbers is always a common multiple, but the smallest one is
the LCM.
For example:
● The multiples of 12 are 12, 24, 36, 48, 60...
● The multiples of 10 are 10, 20, 30, 40, 50, 60...
The highest common factor (HCF) of two numbers is the largest factor that divides both
numbers.
● A common factor is any number that can divide both numbers without leaving a
remainder.
● The number 1 is always a common factor of any two numbers.
● To identify common factors, list all factors of each number and find the ones that
appear in both lists.
For example:
● The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
● The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
● The common factors are 1, 2, 3, and 6.
● The highest common factor is 6.
Finding the HCF Using Prime Factorization
1. Write each number as a product of its prime factors.
2. Identify the prime factors that are common to both numbers.
3. Multiply the common prime factors to find the HCF.
Example:
Find the HCF of 42 and 90.
● Prime factorization:
○ 42 = 2 × 3 × 7
○ 90 = 2 × 3 × 3 × 5
● Common prime factors: 2 and 3
● HCF = 2 × 3 = 6
A Venn diagram can help by placing the common prime factors in the center and the unique
factors in separate circles. The product of the center values gives the HCF.
Lowest Common Multiple (LCM)
The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple
of both.
● A common multiple appears in the times tables of both numbers.
● The product of the two numbers is always a common multiple, but the smallest one is
the LCM.
For example:
● The multiples of 12 are 12, 24, 36, 48, 60...
● The multiples of 10 are 10, 20, 30, 40, 50, 60...
