Lowest Common Multiple (LCM) & Highest Common Factor(HCF) – JSS1 Mathematics Lesson Note (First Term)

The lowest common multiple (LCM) of two numbers is the smallest number that both divide into exactly. The highest common factor (HCF) is the largest number that divides both exactly. Both come from writing each number as a product of prime factors. The LCM combines the prime factors of both numbers, and the HCF keeps only the common ones.

Behavioural objectives

By the end of the lesson, students should be able to:

• state what a multiple, a factor and a prime number are;
• list the factors and factor pairs of a number;
• express a number as a product of prime factors in index form, using repeated division and a factor tree;
• find the LCM of two numbers;
• find the HCF of two numbers.

What are multiples, factors and prime numbers?

A prime number is a number that can only be divided exactly by 1 and itself. It has exactly two factors, which are 1 and itself. Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, etc.

Multiples: A multiple of a number is obtained by multiplying it by any whole number. For example, the multiples of 4 are 4, 8, 12, 16, 20, 24, etc.

Factors: A factor of a number is a whole number that divides the number exactly.

Example 1

(a) Find all the factors of 18.
(b) State which of these factors are even.
(c) State which of these factors are prime numbers.
(d) Write the first three multiples of 18.

Solution

• (a) The factors of 18 are 1, 2, 3, 6, 9 and 18.
• (b) The even factors are 2, 6 and 18.
• (c) The prime factors are 2 and 3.
• (d) The first three multiples of 18 are 18, 36 and 54.

Example 2

Find the factor pairs of 56.

Solution

• 1 × 56
• 2 × 28
• 4 × 14
• 7 × 8

Therefore, the factors of 56 are 1, 2, 4, 7, 8, 14, 28 and 56.

How do you write a number as a product of prime factors?

A prime factor is a factor that is also a prime number. You can find the product of prime factors of a number by the prime factor tree method or by repeatedly dividing by prime numbers.

Example 3

Express 56 and 108 as products of prime factors in index form.

Solution

Method 1: Divide repeatedly by prime numbers.

56 = 2 × 2 × 2 × 7. Index form = 2³ × 7.

108 = 2 × 2 × 3 × 3 × 3. Index form = 2² × 3³.

Method 2: Factor tree.

For 56:

• 56 = 2 × 28
• 28 = 2 × 14
• 14 = 2 × 7

The prime numbers at the ends of the branches are 2, 2, 2 and 7, so 56 = 2³ × 7.

For 108:

• 108 = 2 × 54
• 54 = 2 × 27
• 27 = 3 × 9
• 9 = 3 × 3

The prime numbers at the ends of the branches are 2, 2, 3, 3 and 3, so 108 = 2² × 3³.

Note that every number at the end of a branch must be a prime number.

How do you find the LCM of two numbers?

Write each number as a product of prime factors, then multiply the prime factors together, counting each prime as many times as it appears most often in either number. You can also divide both numbers by prime numbers until you reach 1 in both columns, then multiply all the divisors.

Example 4: Find the LCM of 18 and 24

Method 1: Repeated division.

LCM = 2 × 2 × 2 × 3 × 3 = 72

Method 2: Prime factors.

• 18 = 2 × 3 × 3
• 24 = 2 × 2 × 2 × 3
• LCM = 2 × 2 × 2 × 3 × 3 = 72

Example 5: Find the LCM of 72 and 90

Solution

Using prime factors:

• 72 = 2 × 2 × 2 × 3 × 3
• 90 = 2 × 3 × 3 × 5
• LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360

How do you find the HCF of two numbers?

Write each number as a product of prime factors and pick the common ones. The HCF is the product of the prime factors that both numbers share.

Example 6: Find the HCF of 72 and 96

Solution

• 72 = 2 × 2 × 2 × 3 × 3
• 96 = 2 × 2 × 2 × 2 × 2 × 3
• HCF = 2 × 2 × 2 × 3 = 24

Evaluation

1. Find all the factors of 30 and state which of them are prime numbers.
2. Write the first four multiples of 7.
3. Express 84 as a product of prime factors in index form.
4. Find the LCM of 12 and 18.
5. Find the HCF of 48 and 72.

Assignment

1. Find the factor pairs of 64.
2. Express 90 as a product of prime factors in index form, using a factor tree.
3. Find the LCM of 36 and 60.
4. Find the HCF of 36 and 60.