__Objectives:__

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

- write the highest common factors of a set of given numbers.
- write the highest common factor of a given set of numbers by the use of prime factors or a factor tree.
- the highest common factor also known as the greatest common factor is the biggest factor common to a set of numbers.

__Example: 1__

Write the highest common factor of each of these set of numbers.

- 12 and 18
- 15 and 25

__Solution__

- 12 = {1, 2, 3, 4, 6, 12}

18 = {1, 2, 3, 6, 99, 18}

Common factors {1, 2, 3}

Highest common factor {6}

- 15 = {1, 3, 5, 15}

25 = {1, 5, 25}

Common factors = {1, 5)

Highest common factor = {5}

**From the examples above, all the factors of each number are listed. **

The factors common to both numbers are then listed or selected. After listing the common factors, the biggest one or the highest factor is chosen.

__Example: 2__

Find the highest common prime factors of the following set of numbers.

- 24 and 32 b) 28 and 36

__Solution__

- 24 = Prime factors = 2 x 2 x 2 x 3
- 32 = Prime factor = 2 x 2 x 2 x 2 x 2

Common factors of 24 and 32 = 2 x 2 x 2

Highest common factor = 2^{3} = 8

__Example: 3__

Write the HCF of the following pairs of numbers using the ladder method.

- 30 and 40
- 45 and 65

__Solution__

From the example above, the numbers are divided using a factor that is common to both. The division process is repeated until there are no more common factors. The factors are then listed and multiplied to get the highest common factor.

As in example a, 2 and 5 are common factors. Therefore 2 x 5 = 10 which means that 10 is the highest factor or number that can divide 30 and 40 without a remainder.

__Example: 4__

Finding the HCF of a set of numbers using a factor tree.

With the aid of a factor tree, find the HCF of these set of numbers.

- 15 and 21
- 36 and 60
- 25 and 35

The common prime factors are 2 x 2 x 3 or 2^{2} x 3

Hence, the HCF of 36 and 60 = 2 x 2 x 3 = 12.

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