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 exactly divide into, and the highest common factor (HCF) is the largest number that divides both exactly.

Behavioural objectives

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

1. State what a multiple, a factor and a prime number are.
2. List the factors and factor pairs of a number.
3. Express a number as a product of prime factors in index form, using repeated division and a factor tree.
4. Find the LCM of two numbers.
5. Find the HCF of two numbers.

What are multiples, factors and prime numbers?

A multiple is a number you get by multiplying a given number by a whole number, a factor is a number that divides it exactly, and a prime number has exactly two factors.

• Multiple: a number obtained by multiplying a number by any whole number. The multiples of 4 are 4, 8, 12, 16, 20, 24, etc.
• Factor: a whole number that divides another number exactly, with no remainder.
• Factor pair: two factors whose product is the number. For example, 7 × 8 is a factor pair of 56.
• Prime number: a number that can only be divided exactly by 1 and itself. It has exactly two factors.
• Prime factor: a factor that is also a prime number. For example, 2 and 3 are prime factors of 18.
• Index form: a short way of writing repeated multiplication. 2 × 2 × 2 = 2³, where 2 is the base and 3 is the index.

Facts about prime numbers

• Examples: 2, 3, 5, 7, 11, 13, 17, 19, etc.
• The number 1: it is not a prime number, because it has only one factor.
• The number 2: it is the only even prime number.

Example 1

Answer the following for the number 18.

• (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?

Writing a number as a product of prime factors means expressing it using only prime numbers multiplied together.

There are two methods: repeated division and the factor tree.

Method 1: Repeated division

1. Divide the number by the smallest prime number that divides it exactly.
2. Divide the answer by the smallest prime that divides it exactly.
3. Continue until the answer is 1.
4. Multiply all the divisors together.
5. Write repeated primes in index form.

Method 2: Factor tree

1. Write the number as a product of any two factors.
2. Break each factor that is not prime into two smaller factors.
3. Stop when every number at the end of a branch is prime.
4. Multiply the prime numbers at the ends of the branches.

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

Which prime should you divide by?

• 2: use it when the last digit is 0, 2, 4, 6 or 8.
• 3: use it when the sum of the digits divides by 3. For example, 1 + 0 + 8 = 9, so 108 divides by 3.
• 5: use it when the last digit is 0 or 5.

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.

What is the lowest common multiple (LCM)?

The LCM of two numbers is the smallest number that is a multiple of both numbers.

Finding the LCM by listing multiples

• Multiples of 4: 4, 8, 12, 16, 20, 24, etc.
• Multiples of 6: 6, 12, 18, 24, etc.
• Common multiples: 12, 24, etc.
• LCM: the smallest common multiple, which is 12.

Listing is slow for large numbers, so use prime factors or repeated division.

How do you find the LCM of two numbers?

Write each number as a product of prime factors, then multiply the primes, counting each prime as many times as it appears most often in either number.

Steps: prime factor method

1. Write each number as a product of prime factors.
2. List every prime that appears in either number.
3. Take each prime the greatest number of times it appears in one number.
4. Multiply them together.

Steps: repeated division method

1. Write both numbers side by side.
2. Divide by a prime that divides at least one of them.
3. Copy down any number that the prime does not divide.
4. Continue until both columns reach 1.
5. 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

What is the highest common factor (HCF)?

The HCF of two numbers is the largest number that divides both of them exactly.

Finding the HCF by listing factors

• Factors of 12: 1, 2, 3, 4, 6, 12.
• Factors of 18: 1, 2, 3, 6, 9, 18.
• Common factors: 1, 2, 3, 6.
• HCF: the largest common factor, which is 6.

How do you find the HCF of two numbers?

Write each number as a product of prime factors and multiply the prime factors that both numbers share.

Steps: prime factor method

1. Write each number as a product of prime factors.
2. Pick the primes that appear in both numbers.
3. Take each common prime the smallest number of times it appears in either number.
4. Multiply them together.

Example 6: Find the HCF of 72 and 96

Solution

• 72 = 2 × 2 × 2 × 3 × 3
• 96 = 2 × 2 × 2 × 2 × 2 × 3
• Common primes: three 2s and one 3.
• HCF = 2 × 2 × 2 × 3 = 24

What is the difference between LCM and HCF?

The LCM is the smallest number that both numbers divide into, while the HCF is the largest number that divides into both.

How are LCM and HCF used in problems?

Use the LCM when events repeat and must meet again, and use the HCF when items are shared into the largest equal groups.

Example 7: LCM problem

Two bells ring every 12 minutes and every 15 minutes. If they ring together now, after how many minutes will they next ring together?

• 12 = 2 × 2 × 3
• 15 = 3 × 5
• LCM = 2 × 2 × 3 × 5 = 60
• The bells ring together again after 60 minutes.

Example 8: HCF problem

A trader has 24 oranges and 36 mangoes. She packs them into the largest number of equal groups with nothing left over. How many groups can she make?

• 24 = 2 × 2 × 2 × 3
• 36 = 2 × 2 × 3 × 3
• HCF = 2 × 2 × 3 = 12
• She makes 12 groups, each with 2 oranges and 3 mangoes.

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

Do all the parts below.

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.