Course:
Mathematics Form 1
Numbers are everywhere! Whether you’re counting your change at the store, 🛒 checking the animals on the farm 🐄, or celebrating another birthday 🎉, numbers play a huge role in our lives. But did you know that numbers come in different “classes”?
In this lesson, we’ll explore the amazing world of numbers! We’ll start with the basics—odd and even numbers—and then dive into fascinating types like prime numbers, triangle numbers, cube numbers, and more. Get ready to discover the secrets hidden in these magical digits! ✨
By the end of this lesson, you’ll be able to:
We’ll review the key points to help you master whole numbers and get ready for more challenges ahead. 🏆
Test your skills with our Self-Assessment Exercises. Are you ready to show off what you’ve learned? Let’s go! 🚀💥
As I have already mentioned, there are different ways of classifying whole numbers. Whole numbers are also called integers. Integers include negative whole numbers ( e.g –3, -5, -49, e.t.c. ), positive whole numbers ( e.g. 1,2,3,4,5,e.t.c. ) and the number zero. Positive whole numbers are also called counting numbers or natural numbers. We can split the counting numbers / positive whole numbers into two main classes, namely; Odd numbers and Even numbers.
These are numbers which always leave a remainder when divided by 2
e.g.📄 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, etc.
These are numbers which can be divided by 2 without leaving a remainder
e.g.📄 2, 4, 6, 8, 10,12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, e.t.c.
A prime number is a number which has only two factors, the two factors are 1 and itself. A factor of a number is a number that can divide that number without leaving a remainder. Examples of Prime Factors include the following;
2,3,5,7,11,13,17,19,23,29,31,37,41,43,e.t.c
From the list above we can see that;
2 has only two factors 1 and 2
3 has only two factors 1 and 3
5 has only two factors 1 and 5
7 has only two factors 1 and 7
and so on.
What are triangle numbers? ✨
These are numbers that can be arranged in the shape of a triangle, as illustrated below:
What are rectangle numbers?✨
These are numbers that can be arranged in the shape of a rectangle, as illustrated below:
Note that the rectangles for 6, 8, 10, 12 and so forth can also be horizontal as shown below;
What are square numbers?✨
These are numbers that are formed when multiplying a whole number by itself.
For example 📝
| 12 = 1 x 1 = 1, therefore 1 is a square number 22 = 2 x 2 = 4, therefore 4 is a square number 32 = 3 x 3 = 9, therefore 9 is a square number 42 = 4 x 4 = 16, therefore 16 is a square number etc. |
What are cube numbers? ✨
These are numbers that are formed when a whole number is multiplied by itself three times.
For example 📝
|
13 = 1 x 1 x 1 = 1, therefore 1 is a cube number 23 = 2 x 2 x 2 = 8, therefore 8 is a cube number 33 = 3 x 3 x 3 = 27, therefore 27 is a cube number 43 = 4 x 4 x 4 = 64, therefore 64 is a cube number 53 = 5 x 5 x 5 = 125, therefore 125 is a cube number |
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
As already discussed, there are two main classes of whole numbers, namely odd numbers and even numbers. These numbers have factors. A factor of any given number is a number that can divide that given number without leaving a remainder.
Example 1 📝
(a) 15 is an odd number. It can be divided by 1, 3, 5 and 15 without leaving a remainder, hence the numbers 1,3,5 and 15 are called factors of 15.
(b) 24 is an even number. It has factors 1, 2, 3, 4, 6, 8, 12 and 24.
(c) 73 is both an odd number and a prime number. It has factors 1 and 73 only.
Sometimes it is easy to leave out some factors of a number. The best way to list without leaving out is to use the method of pairing factors, as shown in example 2.
Example 2 📝
(a) List factors of 80
(b) List factors of 49
Solution ✅
(a) i)
80 ÷ 1 = 80. The first two factors of 80 are 1 and 80 because 80 is divisible by both 1 & 80. Similarly, when multiplying the pair of factors, you get 80,
that is, 1 x 80 = 80
(ii) 80 ÷ 2 = 40. The next two factors of 80 are 2 & 40. Again when you multiply this pair of factors, you get: 2 x 40 = 80
(iii) 80 ÷ 4 = 20. The next two factors of 80 are 4 and 20.
(iv) 80 ÷ 5 = 16. The next pair of factors are 5 and 16.
(v) 80 ÷ 8 = 10. The next are 8 and 10.
All of these can be represented in a small table as follows,
| 1 | 80 |
| 2 | 40 |
| 4 | 20 |
| 5 | 16 |
| 8 | 10 |
Therefore factors of 80 are 1, 80, 2, 40, 4, 20, 5, 16, 8, 10 and they can be arranged in order of size, starting with the smallest as shown below:
1, 2, 4, 5, 8, 10, 16, 20, 40, 80.
1 is the lowest factor of 80 and 80 is the highest factor of 80.
(b) (i) 49 ÷ 1 = 49. The factors here are 1 and 49.
(ii) 49÷7=7,thefactorshereare7and7.Butwhenwelist,welistonlyone7.
Therefore, factors of 49 are: 1, 7, 49.
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Let us consider factors of 30 and 24.
Factors of 30 are; 1,2,3,5,6,10,15,30
Factors of 24 are; 1,2,3,4,6,8,12,24
Some of the factors of 30 are also factors of 24. These are 1,2,3 and 6, and they are called common factors of 30 and 24. When you look at these common factors, you will realize that the highest is 6 and it is therefore called the highest common factor (H.C.F) of 30 and 24.
Example 3 📝
What is the highest common factor in each case?
(a) 25 and 35
(b) 20 and 100
Solution ✅
(a) (i) Factors of 25 are: 1, 5, 25
(ii) Factors of 35 are: 1, 5, 7,35
The common factors of 25 and 35 are 1 and 5, and as such the H. C. F. of 25 and 35 is 5.
(b) (i) Factors of 20 are: 1, 2, 4, 5, 10, 20
(ii) Factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, 100
The common factors of 20 and 100 are: 1,2,4,5,10,20. Hence the HCF is 20.
When listing factors of numbers, we listed factors of 15 as 1,3,5,15. These are numbers that can divide 15 without leaving a remainder. There are some other numbers which can be divided by 15, without leaving a remainder. For example, 30 is divisible by 15 and 90 is also divisible by 15. These numbers are therefore called multiples of 15, and they are many.
Multiples may also be obtained by multiplying a number by other counting numbers as shown below;
Multiples of 6 are:
6×1= 6
6×2= 12
6×3= 18
6×4= 24
and so on….
Therefore multiples of 6 are 6,12,18,24,…
A NUMBER IS A MULTIPLE OF ITSELF.
Example 4 📝
List multiples of the following numbers:
(a) 15
(b) 24
(c) 17
Solution: ✅
(a) Multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, etc.
Note that 15 is the lowest multiple of 15.
(b) Multiples of 24 are: 24, 48, 72, 96, 120, 144, etc.
(c) Multiples of 17 are: 17, 34, 51, 68, 85, etc.
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The Lowest Common Multiple (L.C.M.)
To find the Lowest Common Multiple of two or more numbers, first list the multiples of each number. Then list all those that appear in all the sets then pick the lowest in that set. Let us start with 2 and 5.
Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, etc.
Multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.
Some multiples of 5 are also multiples of 2. These are 10, 20, 30, 40, 50, etc. These are called common multiples of 5 and 2, and the lowest is 10, hence it is called the lowest common multiple (L.C.M.) of 5 and 2.
Example 5 📝
What is the L.C.M. of the following numbers?
(a) 4 and 6.
(b) 50 and 200
Solution ✅
(a) Multiples of 4 are: 4,8,12,16,…
Multiples of 6 are 6,12,18,24,…
Hence the L.C.M. of 4 and 6 is 12
(b) Therefore L.C.M. of 50 and 200 is 200
Multiples of 50 are 50,100,150,200,250….
Multiples of 50 are 200, 400, 600….
We have already listed factors of numbers in pairs. For example, factors of 6 can be written as 1 and 6, 2 and 3. Each pair of factors can be multiplied to give the number 6 as follows: 1 x 6 = 6 and 2 x 3 = 6. Remember the word product? It involves multiplication, the product of 2 and 3 is 6.
We can therefore, express a number as a product of its factors.
Example 6 📝
Express the following numbers as products of their factors:
(a) 28
(b) 100
(c) 35
Solution ✅
(a) Factors of 28 in pairs are: 1 and 28, 2 and 14, 4 and 7. Therefore, 28 as a product of its factors is:
1 x 28
or 2 x 14
or 4 x 7.
(b) Factors of 100 in pairs are:
1 and 100,
2 and 50,
4 and 25,
5 and 20,
10 and 10.
Therefore, 100 as a product of its factors is either:
1 x 100,
or 2 x 50 or
4 x 25 or
5 x 20
(c) 35 as a product of its factors is:
1 x 35
or 5 x 7
The number 42 is a whole number. It has factors:
1,2,3,6,7,14,21,42. Some of these factors are odd numbers ( 1,3,7,21 ). Some are even numbers ( 2,6,14,42 ), and some are prime numbers ( 2,3,7 ).
The odd ones are called odd factors of 42, the even ones are called even factors of 42 and the prime ones are called prime factors of 42.
Example 7 📝
List the odd factors,even factors, and prime factors of each number:
(a) 110
(b) 30
(c) 60
Solution ✅
(a) Factors of 110 are: 1,2,5,10,11,22,55,110 Look at the list and you will notice that the:
(i) Odd factors of 110 are: 1,5,11,55
(ii) Even factors of 110 are: 2,10,22,110
(iii) Prime Factors of 110 are 2,5,11
(b) Factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
(i) Odd factors of 30 are: 1, 3, 5, 15
(ii) Even factors of 30 are: 2, 6, 10, 30
(iii) Prime factors of 30 are: 2, 3, 5
(c)Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12,1 5, 20, 30, 60
(i) Odd factors of 60 are 1, 3, 5, 15
(ii) Even factors of 60 are: 2, 4, 6, 10, 12, 20, 30, 60
(iii) Prime factors of 60 are: 2, 3, 5
Since we have already expressed numbers as products of their factors, now we are going to express them as products of their prime factors.. In Activity 8, we expressed 70 as a product of its factors as 1×70, or 2×35 or 5×14 or 7×10.
Now, 70 as a product of its prime factors is 2x5x7
Example 8 📝
Look at Example 7. The prime factors of 110 are 2,5 and 11. Therefore, 110 as a product of its prime factors is 2x5x11. The following two methods are used to help us find prime factors of numbers.
Express each number as a product of its prime factors:
(a) 30
(b) 210
Solution ✅
(a) (i) Method 1
| 2 | 30 |
We divide 30 by the first prime number (2), and then the next prime number and so on until we get 1 as the last number. |
| 3 | 15 | |
| 5 | 5 | |
| 1 |
Therefore, 30 as a product of its prime factors is 30 = 2x3x5
(ii) Method 2
In this method, think of the smallest prime number that divides 30, that is 2. Therefore,
30 = 2 x 15, which is the second row. Again think of the smallest prime number that divides 15, that is 3. Therefore, 15 = 3 x 5, which is the thid row.
Hence 30 = 2 x 3 x 5 We begin by dividing 210 by the smallest prime number 2 and continue until we get 1 as the last number.
(b) (i) Method 1
| 2 | 210 |
We begin by dividing 210 by the smallest prime number 2 and continue until we get 1 as the last number. |
| 3 | 105 | |
| 5 | 35 | |
| 7 | 7 | |
| 1 |
Therefore 210 as a product of its prime factors is 210 = 2x3x5x7
(ii) Method 2
Hence 210 = 2x3x5x7
As you can see the methods are basically the same so it is up to you to choose the one you are more comfortable with. Now try the exercise below.
(1) List all the prime factors of each number;
(a) 28
(b) 400
(c) 184
The following are some of the key points that we need to remember;
A number can either be odd or even.
All prime numbers are odd, except the number 2, but not all odd numbers are prime numbers.
All prime numbers are not rectangle numbers.
The number 2 is a prime number and it is also an even number.
Factors of a number should divide into the number without leaving a remainder
Multiples of a number should be divisible by the number without leaving a remainder
H.C.F. is short for Highest Common Factor
L.C.M. is short for Lowest Common Multiple
Expressing a number as a product of its factors means multiplying any pair of factors of that number to give the number.
Expressing a number as a product of its prime factors means multiplying only the prime factors of that number to give the number.
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