3. Multiples
A multiple of a number is found by multiplying the number by any whole number.
What are the first six multiples of 7?
To find the first six multiples of 7, multiply 7 by 1, 2, 3, 4, 5 and 6 in turn.
7 × 1
7 × 2
7 × 3
7 × 4
7 × 5
7 × 6
Teacher notes
Discuss the fact that any given number has infinitely many multiples.
We can check whether a number is a multiple of another number by using divisibility tests. 7 14 21 28 35 42
Any given number has infinitely many multiples.

4. Multiples

5. Ordering multiples

7. Factors
A factor (or divisor) of a number is a whole number that divides into it exactly.
Factors come in pairs.
What are the factors of 30?
1 and 30,
2 and 15,
3 and 10,
5 and 6.
Teacher notes
Ask pupils to tell you what a factor is and reveal the definition on the board.
Remind pupils that factors always go in pairs (in the example of rectangular arrangements these are given by the length and the width of the rectangle). The pairs multiply together to give the number.
Ask pupils if numbers always have an even number of factors. They may argue that they will, because factors can always be written in pairs. Establish, however, that when a number is multiplied by itself the numbers in that factor pair are repeated. That number will therefore have an odd number of factors. Pupils could investigate this individually.
Establish that it follows that if a number has an odd number of factors, it must be a square number, e.g. 25 has factors of 1, 5 and 25.
In ascending numerical order, the factors of 30 are: 1, 2, 3, 5, 6,
10, 15
and 30.

8. Factor recognition

9. Floor tiles This pattern from a kitchen floor shows 9 floor tiles.
The only way they can be arranged is:
in a rectangle that is 1 tile × 9 tiles
in a square that is 3 tiles × 3 tiles.
How many ways can 18 tiles be arranged?
What is special about numbers that can be arranged into squares?
What is special about numbers that can only be arranged into rectangles that are one tile wide?
Teacher notes
18 tiles could be arranged in: a rectangle 1 tile long × 18 tiles wide; a rectangle 2 tiles long × 9 tiles wide; a rectangle 3 tiles long × 6 tiles wide; a rectangle 6 tiles long × 3 tiles wide and one that is a rectangle 9 tiles long × 2 tiles wide.
Numbers that can only be arranged into squares are square numbers. This means that the numbers have an odd number of factors as opposed to an even number. For example, if you have 36 tiles, they could be arranged: 1 tile × 36 tiles; 2 tiles × 18 tiles; 3 tiles × 12 tiles; 4 tiles × 9 tiles; 6 tiles × 6 tiles and vice versa. The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36. This means that there are 9 factors. This is because we multiply the number 6 by itself to give us 36. This is true of all square numbers: one factor multiplied by itself will give us a square number.
Numbers that can only be arranged into rectangles that are one tile wide are known as prime numbers. These numbers do not divide into any numbers other than themselves and one. For example, if you had 11 tiles, you would only be able to arrange them in a rectangle that was 1 tile × 11 tiles.
Encourage students to do their own investigations with tile patterns and numbers. You could cut up squares of cards into tiles and get the students to create different arrangements for different numbers.

11. All your eggs in one basket
Sonia and Shaun help their parents run an organic food store.
Mr Jones gets a delivery of 12 eggs every week.
Mr Smith gets a delivery of 13 eggs every week.
To protect the eggs, they are delivered in rectangular boxes containing the exact number of eggs.
Teacher notes
Mr Jones gets a delivery of 12 eggs every week. You could fit his eggs into several different shaped boxes. You could fit them into a box that was one egg wide and 12 eggs long; you could fit them in a box that was 2 wide and 6 long or you could fit them in a box that was 3 wide and 4 long.
On the other hand, Mr Smith’s delivery could only fit in one type of box.13 is a prime number, so you could only fit 13 eggs in a box that was one egg wide and 13 eggs long.
You could encourage students to design a suitable box for the eggs to introduce an investigative element to this slide.
What size of box could be used for Mr Jones’ eggs?
What size of box could be used for Mr Smith’s eggs?

12. Prime numbers
If a whole number has two, and only two, factors, it is called a prime number.
Can you think of any prime numbers?
Which of the numbers in this list is a prime number? 1
1021 9 81 6 52
100 17
Teacher notes
The number 17 has only two factors, 1 and 17. Therefore, 17 is a prime number.
The other numbers in the list have at least 3 factors (except the number 1), so they cannot be prime numbers. Encourage the students to write down all the factors of the numbers listed on the slide.
The number 1 has only one factor: 1. Therefore, 1 is not a prime number.
Establish that 2 is the only even prime number.
Ask pupils to name all the prime numbers less than 20. Ask them to list all the prime numbers less than 100. Is there an easy method of doing this?
Why is the number 1 not a prime number?
There is only one even prime number. What is it?

13. Prime numbers
Eratosthenes of Cyrene was a mathematician who lived from 276 – 194 BC.
He was responsible for developing a method of identifying prime numbers up to a given number.
This method is known as the Sieve of Eratosthenes.
Teacher notes
Ask students to use their research skills to find out more about the Sieve of Eratosthenes. Some useful websites to look at may include:
They will get a chance to practice using the Sieve of Eratosthenes to identify all the prime numbers up to 100 by completing the activity on the following slide.
Ask pupils to learn the first 10 prime numbers. They will be better prepared for their exam if they can easily identify the first 10 prime numbers.
What do you know about the Sieve of Eratosthenes?
How can you use it to tell if a number is prime?

14. Sieve of Eratosthenes Teacher notes
Start by defining a prime number as a number that has only two factors: the number one and the number itself. Demonstrate how to find the prime numbers less than 100 using the Sieve of Eratosthenes.
Start by selecting the number 1 and colouring it yellow. Explain that this is not a prime number because it only has one factor.
We then need to colour all the multiples of 2 not including the number 2 itself. This can be done quickly by selecting all of the multiples of two from the menu at the bottom of the screen and colouring them. 2 is a prime number and should not be coloured. It is necessary to clear the current selection, select the 2 and colour it white.
Repeat this process for multiples of 3, not forgetting to colour it white after shading its multiples yellow.
Proceed to the next uncoloured prime number, which is 5. Colour multiples of 5 and then 7, not including 5 and 7 themselves, to complete the activity.
The next uncoloured number is 11. Ask pupils why we do not need to colour multiples of 11 or any other prime number on the square. Verify that we have found all the prime numbers less than 100 by selecting prime numbers from the menu at the bottom of the screen and showing that they coincide with the uncoloured numbers in the number square.

15. Online transactions Rose uses her home computer to download music.
Her mum uses the computer to pay bills online.
Her brother shops online for his computer games.
How do you think maths is used when buying things over the Internet?
How can prime numbers help you tell the seller your bank details, but keep them secret from everyone else?
Teacher notes
Prime numbers are used as the basis for encryption purposes when online transactions are being conducted. The seller will take two huge prime numbers, a and b (these may be 100 to 200 digits long), and multiply them together to give a product that is divisible by one, itself and by the two original prime numbers. Let’s call this number, N. To find the two factors of N will be incredibly difficult.
The prime numbers are then used within an algorithm known as the RSA algorithm to encrypt the data. They are then also needed to decrypt the data.
You may find this theory easiest to explain to your students by way of an analogy. Suppose I send you a box with an open padlock in the mail but hold on to the key. You put your secret information in the box, close the padlock and return it to me. I am now the only person who can access your information as I am the only person with the right key to unlock the box.
For further information of prime numbers and encryption, see:

17. Prime factors
A prime factor is a factor of a number that is a prime number.
What are the prime factors of 70?
The factors of 70 are: 1 2 5 7 10 14 35 70
Teacher notes
Ask for the factors of 70 before revealing them.
Then ask which of these factors are prime numbers.
Ask the students to write down all the factors of 52 before noting which of these factors are prime. This will help ensure none of the factors are missed: 1, 2, 4, 13, 26 and 52. Of these factors, 2 and 13 are prime factors.
The prime factors of 70 are: 2, 5, and 7.
What are the prime factors of 52?

18. Identifying prime factors

19. Products of prime factors
A whole number can be written as a product of prime factors. 70
= 2 × 5 × 7 56
= 2 × 2 × 2 × 7
= 2³ × 7 99
= 3 × 3 × 11
= 3² × 11
If we were to try this with any whole number greater than 1, what do you think the outcome would be?
Every whole number greater than 1 can be written as either:
a prime number
a product of two or more prime numbers.
Teacher notes
First of all, explain that one of the reasons prime numbers are so important is that by multiplying together prime numbers you can make any whole number bigger than one.
Go through each of the products. Remind pupils of index notation and how to read it. For example, 22 is read as “2 squared” or “2 to the power of 2”. 23 is read as “2 cubed” or “2 to the power of 3”
Tell pupils that every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers.
This is called the Fundamental Theorem of Arithmetic. Pupils are not expected to know this term, but it is an important idea. It is a good reason for defining prime numbers to exclude 1. If 1 were a prime, then the prime factor decomposition would lose its uniqueness. This is because we could multiply by 1 as many times as we liked in the decomposition.
Numbers that are not prime are often called composite numbers because they are made up of products of prime numbers.
Try it yourself! Can you find any exceptions to the rule?

20. The prime factor decomposition
Writing a number as a product of prime factors is called the prime factor decomposition or prime factor form.
The prime factor decomposition of 100 is:
100 = 2 × 2 × 5 × 5
= 22 × 52
Teacher notes
Verify that 2 × 2 × 5 × 5 = 100.
Encourage students to think of what the two main methods of finding the prime factor decomposition of a number are. They should have encountered these methods at KS3 level, but may need reminding about how the two different methods work. A detailed explanation and examples are shown on the following slides.
There are two main methods of finding the prime factor decomposition of a number.

21. Prime factor decomposition
Teacher notes
Factor tree method: explain that to write 36 as a product or prime factors we start by writing 36 at the top (of the tree). Next, we need to think of two numbers which multiply together to give 36: in this example we are using 9 and 4. Explain that it doesn’t matter whether we use: 2 × 18, 3 × 12, 4 × 9, or 6 × 6, the end result will be the same.
Next, we must find two numbers that multiply together to make 4: 2 is a prime number so we can stop there. Now find two numbers which multiply together to make 9: 3 is a prime number so we can stop there.
State that when every number at the bottom of each branch is circled we can write down the prime factor decomposition of the number writing the prime numbers in order from smallest to biggest.
Explain that it’s usual to use index notation for this, particularly with large numbers.
Repeated division by prime factors: start by writing 36 in the table. What is the lowest prime number that divides into 36? Write the 2 to the left of the 36 and then divide 36 by 2. We write the result of this division under the 36. What is the lowest prime number that divides into 18? Continue dividing by the lowest prime number possible until you get to 1. When you get to 1 at the bottom, stop.
The prime factor decomposition is found by multiplying together all the numbers in the left hand column. Again, this is usually represented using index notation.

22. Factor trees
How would you write the prime factor decomposition of 2100 using a factor tree?
2100 70 30 6 5 10 7
Teacher notes
Explain that there are many ways to draw the factor tree for 2100 but the final factor decomposition will be the same.
The prime factors are written in order of size and then simplified using index notation.
Encourage students to complete the prime factor tree in a different way to try and disprove the theory. They will always come up with exactly the same decomposition, no matter how they approach the factors. 2 3 2 5
2100 = 2 × 2 × 3 × 5 × 5 × 7
= 22 × 3 × 52 × 7

23. Factor trees Teacher notes
This activity provides another two examples of using the factor tree method of prime factor decomposition. Should the students prefer using the repeated division by prime numbers method, there is more practice on the following slides. Ideally, students would be able to move between the two different methods. However, when going through this activity, feel free to get students to conduct the decomposition in their own way. The end result, if performed correctly, is exactly the same.

24. Prime factor decomposition
Use the tree method to find the prime factor decomposition of the following numbers: 160, 240 and 192.
Take two trees at a time and compare their prime factors.
How many of the prime factors are the same in each tree?
Multiply the prime factors. What do you notice about the number that you get as a result?
Teacher notes
The number 160 when written as a prime factor decomposition is: 2 × 2 × 2 × 2 × 2 × 5 = 25 × 5.
The number 240 when written as a prime factor decomposition is: 2 × 2 × 2 × 2 × 3 × 5 = 24 × 3 × 5.
The number 192 when written as a prime factor decomposition is: 2 × 2 × 2 × 2 × 2 × 2 × 3 = 26 × 3.
When you take two of the trees at a time, you will notice that they share certain prime factors.
When you multiply the prime factors that are common to the two trees, you end up with a number that goes into both numbers.
For example: the numbers 160 and 240 have a common factor of 24 × 5 = 80.
160 ÷ 80 = 2.
240 ÷ 80 = 3.

25. Dividing by prime numbers
How would you write the prime factor decomposition of 96 using repeated division by prime numbers? 2 96
Write 96 in the right-hand column of the table. What is the lowest prime number that divides into this? 2 48 2 24
Continue dividing the number in the right-hand column by the lowest prime number possible until you end up with 1 in the right column. 2 12 2
Teacher notes
Explain that another method to find the prime factor decomposition is to divide repeatedly by prime factors putting the answers in a table as follows:
To find the prime factor decomposition of 96 start by writing 96. Click to reveal 96.
Now, what is the lowest prime number that divides into 96? Establish that this is 2. Remind pupils of tests for divisibility if necessary. Any number ending in 0, 2, 4, 6, or 8 is divisible by 2.
Write the 2 to the left of the 96 and then divide 96 by 2. Click to reveal the 2.
This may be divided mentally. Discuss strategies such as halving 90 to get 45 and halving 6 to get 3 and adding 45 and 3 together to get 48.
We write this under the 96.
Now, what is the lowest prime number that divides into 48? Establish that this is 2 again. Continue dividing by the lowest prime number possible until you get to 1.
When you get to 1 at the bottom, stop.
The prime factor decomposition is found by multiplying together all the numbers in the left hand column. 6 3 3 1
96 = 2 × 2 × 2 × 2 × 2 × 3
= 25 × 3

26. Dividing by prime numbers
Teacher notes
Talk through these examples, as before. Remind pupils that to test for divisibility by 3 we must add together the digits and check whether the result is divisible by 3.
The students will need to enter the correct answers in the correct space. Encourage the students to remember that we need to do repeated division by the lowest prime number at each stage in order to correctly complete the prime factor decomposition tables.

28. Using the prime factor decomposition
Using the prime factor form of a number can help us to work out if a number has any special properties. 2
324
324 = 2 × 2 × 3 × 3 × 3 × 3
= 22 × 34 2
162
This could also be written as: 3 81
(2 × 32) × (2 × 32)
or (2 × 32)2. 3 27 3 9
Teacher notes
If all the indices in the prime factor decomposition of a number are even, this tells us that the number is a square number.
If necessary break this down further to show that 324 = (2 × 3 × 3) × (2 × 3 × 3). Ask pupils to use the information on the board to tell you the square root of 324: 324 = 2 × 32 = 2 × 9 = 18.
Verify this by getting students to multiply 18 × 18 either mentally or using a calculator.
If all the indices in the prime factor decomposition of a number are even, what does this tell us about the number? 3 3 1

29. Using the prime factor decomposition
Using the prime factor form of a number can help us to work out if a number has any special properties. 3
3375
3375 = 3 × 3 × 3 × 5 × 5 × 5
= 33 × 53 3
1125 3
375
This could also be written as:
(3 × 5) × (3 × 5) × (3 × 5)
or (3 × 5)3. 5
125 5 25
Teacher notes
If all the indices in the prime factor decomposition of a number are multiples of 3, this tells us that the number is a cube number.
Ask pupils to use the information on the board to tell you the cube root of 3375: 33375 = 3 × 5 = 15.
If all the indices in the prime factor decomposition of a number are multiples of 3, what does this tell you about the number? 5 5 1

30. Using prime factor decomposition

31. Simplifying your calculations
Teacher notes
Q1: 168 × 294 = (23 × 3 × 7) × (2 × 3 × 72 ) = 23 × 2 × 3 × 3 × 7 × 72 = 24 × 32 × 73 (add the indices)
Q2: 4116 ÷ 294 = 22 × 3 × 73 ÷ 2 × 3 × 72 = 2 × 7 = 14 (subtract the indices)
Q3: If 4116 were exactly divisible by 168 then the prime factor decomposition of 168 would not contain prime numbers raised to higher powers than those in the prime factor decomposition of is not divisible by 168 because the prime factor decomposition contains 23, while the prime factor decomposition of 4116 contains 22.
Q4: All the indices in the prime factor decomposition are even: 2 × 3 × 7 × 7 × 2 × 3 = (2 × 3 × 7)². Ask pupils to use this to tell you the square root of 294 × 6.
2 × 3 × 7 = ² = 294 × 6.
Q5: Expressing the fraction using the prime factor decomposition allows you to cancel common factors, leaving you with 4/7.

32. Using prime factor decomposition
Lottery jackpot prizes are quite often claimed by syndicates.
A syndicate is a group of people joined together in a common goal or purpose. In this case, to win the lottery!
45 people won £5680 last week.
92 people won £4140 on the lottery three weeks ago.
Teacher notes
In order to work out whether these amounts could have been split equally into whole pound amounts, we need to divide the amount won by the number of people in the syndicate. Without using a calculator this can be quite difficult, especially when dealing with large numbers. However, by conducting a prime factor decomposition, we should be able to quickly work out an answer.
45 = 3 × 3 × 5 = 3² × 5; 5680 = 2 × 2 × 2 × 2 × 5 × 71 = 24 × 5 × 71; ÷ 45 = 24 × 5 × 71 ÷ 3² × 5
The prime factor decomposition of 5680 contains the prime number 71. This is not divisible by any numbers other than 1 and 71. Therefore, it cannot be divided by 3² × 5. This amount of money can not be split evenly in pounds.
92 = 2 × 2 × 23 = 2² × 23; 4140 = 2 × 2 × 3 × 3 × 5 × 23 = 2² × 3² × 5 × 23; ÷ 92 = 2² × 3² × 5 × 23 ÷ 2² × 23 = 3² × 5 = £45.
Each person in this syndicate got £45.
12 = 2 × 2 × 3 = 2² × 3; 408 = 2 × 2 × 2 × 3 × 17 = 2³ × 3 × 17; 408 ÷ 12 = 2³ × 3 × 17 ÷ 2² × 3 = 2 × 17 = £34.
Each person in this syndicate won £34.
You could further extend this example by giving the students some recent real examples with much larger sums of money. How much did the people involved win?
£408 was recently shared amongst 12 winners.
Were they able to split the money evenly in pounds?