Mathematics

Key Unit Competence
A learner should be able to classify numbers and appreciate that one number 

may belong to various families of numbers.

Attitude and Values
Appreciate the importance of using square roots, being cooperative and 

displaying a teamwork spirit.

3.1 Natural and whole Numbers 
The counting numbers starting from 1 are called natural numbers. 
Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...}
Whole Numbers
Natural numbers together with 0 are called whole numbers. 

Whole numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}

3.2 Odd Numbers

Odd numbers are natural numbers which are not exactly divisible by 2. When 
divided by 2 it always has a remainder of 1.
The following are examples of odd numbers less than 20.

Odd numbers = {1, 3, 5 ,7, 9, 11, 13, 15, 17, 19}

3.3 Even Numbers

Even numbers are natural numbers which are exactly divisible by 2. When 
divided by 2 it leaves no remainder.
The following are the even numbers found between 0 and 20.
Even numbers = {2, 4, 6, 8, 10, 12, 14, 16, 18}.
[Note: 20 is not included in this set of numbers and yet it is an even number. 
This is because we were asked to list even numbers between 0 and 20 and not 

0 to 20.]Activity 3.1
In this class activity, you are going to play 
a game which will help you to learn how 
to classify numbers as whole, natural, odd 
and even numbers.
• The teacher will give you a flashcard 
with a number written on it.
• Look at your number carefully and 
remember it throughout this activity 
(game). 
• When the teacher calls out the class to which your number belongs, 
you will run and stand on the line drawn: Class of odd numbers, even 

numbers, whole numbers and natural numbers.

Exercise 3.1
1. List all the odd numbers between 0 and 100. 
2. List all the even numbers from 0 up to 100.

Think!!!
I am an odd number. Take away one letter and I become even. What number am I?

3.4 Square Numbers
This is the number obtained when one number is multiplied by itself.
Study the multiplication table below and use it to identify all the square numbers 

less than 60.

From the table we can see that;
1 × 1 = 1                               6 × 6 = 36
2 × 2 = 4                               7 × 7 = 49
3 × 3 = 9                               8 × 8 = 64
4 × 4 = 16                            9 × 9 = 81

5 × 5 = 25                           10 × 10 = 100

So the set of square numbers up to 100 = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.

Activity 3.2

Work together in groups and list all the square numbers between 0 and 410.

3.5 Square root of a number
Finding the square root of a number is the inverse operation of squaring the 
number.

Exercise 3.2

Exercise 3.3
Complete the multiplication table below and use it to list all the square 

numbers less than 100.

3.6 Prime Numbers
A prime number is any number with only two factors; one factor being 1 and the 
other one being itself.

Examples of prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

Activity 3.3

List all the prime numbers between 100 and 150.

3.7 Composite Numbers
This is a natural number greater than one which has more than two factors. 
In fact a composite number is a natural number greater than 1 which is not a 
prime number.
The following are examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, ...

Exercise 3.4
1. What is the smallest prime number?
2. What is the smallest even number which is a prime number?
3. What is the smallest odd number which is prime?

4. List the first 10 composite numbers

Mind Game
I am an even number. I am a counting number and I am a prime number too.

Who am I and what is my square?

3.8 Multiples of a Number
The multiples of a whole number are found by taking the product of any counting 

number and that whole number.

For example:

So, the multiples of 5 are 5, 10,15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ...

Exercise 3.5
List the first 10 multiples of the following numbers: 
(a) 4              (b) 6             (c) 7           (d) 8                   (e) 9
(f) 11            (g) 12

3.9 Factors of a Number
The factors of a number are all those whole numbers that can divide evenly to 
the number and leave no remainder. 
The greatest factor of a number is the number itself and the smallest factor of 
a number is 1.

Activity 3.4
Work together in groups and list all the numbers between 1 and 30. Which 
numbers are not multiple of 3?

Example 3.1
List all the factors of 18.

Solution
• We already have two factors of 18 namely 1 and 18. 1 is the smallest 
factor and 18 is the biggest factor. 
• Try dividing 18 by numbers 2, 3, 4, 5 ...
• 18 ÷ 2 = 9 and so 2 is the second smallest factor and 9 is the second 
biggest factor.

• Continue dividing with different numbers until you get all the factors.

• So, the factors of 18 arranged in ascending order are 1, 2, 3, 6, 9, 18.

Exercise 3.6
1. List all the factors of the following numbers:
(a) 8            (b) 12            (c) 24           (d) 36          (e) 48
(f) 64          (g) 45            23           (i) 96             (j) 100
2. I think of a number. When I multiply it by itself, the answer is 100. What 

is the number?

3.10 Lowest Common Multiple (LCM)
LCM means the Lowest Common Multiple. So, the LCM of two numbers is the 
smallest multiple which is common to both numbers.

Example 3.2
Find the LCM of the following numbers:
(a) 3 and 4                         (b) 5 and 12
(c) 12 and 15                    (d) 20 and 65

Solution
Method 1
(a) LCM of 3 and 4
Multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, ...}
Multiples of 4 = {4, 8, 12, 16, 20, 24, 28…}
The common multiples are 12, 24, ........

The smallest common multiple is 12 and so the LCM of 3 and 4 = 12.

(b) LCM of 5 and 12
Multiples of 5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ...}
Multiples of 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, ...}
The lowest common multiple is 60.
So, the LCM of 5 and 12 = 60.

(c) LCM of 12 and 15
Multiples of 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, ...}
Multiples of 15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, ...}
The lowest common multiple is 60.
So, the LCM of 12 and 15 = 60

(d) LCM of 20 and 65
Multiples of 20 = {20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 
240, 260, ...}
Multiples of 65 = {65, 130, 195, 260, 325, 390, ...}
The lowest common multiple is 260.
So, the LCM of 20 and 65 = 260.

Method 2
Another way of finding the LCM of two or more numbers is to first express 
them in terms of prime factors. Let’s use this method to find the LCM of 12 
and 15 and see whether we shall get 60.

Solution

Factorise 12 and 15 in terms of prime factors

The LCM can be got by multiplying all the prime factors of 12 and 15 in the 
first column on the left.

So, LCM of 12 and 15 = 2 × 2 × 3 × 5 = 60.

Assessment Exercise

1. Evaluate the following by finding its square root:

2. Find the first five multiple of the following numbers: 3, 6, 5, 9, 10
3. List the first 10 prime numbers.
 (a) Circle all the square numbers.
 (b) Using a pencil, tick all the composite numbers.
4. Find the highest common factor (HCF) and LCM of the following 
numbers:
 (a) 2 and 4         (b) 4 and 5      (c) 3 and 6     (d) 4 and 10

 (e) 12 and 14     (f) 23 and 46

Internet Resource
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