1. N7 Prime factor decomposition, HCF and LCM
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
N7 Prime factor decomposition, HCF and LCM
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2. N7.1 Prime factor decomposition
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
N7.1 Prime factor decomposition

3. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Prime factors
A prime factor is a factor that is also a prime number.
What are the factors of 30?
The factors of 30 are: 1, 2, 3, 5, 6,
10,
15,
30.
Ask for the factors of 30 before revealing them.
Then ask which of these factors are prime numbers.
The prime factors of 30 are 2, 3, and 5.

4. Products of prime factors
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Products of prime factors
2 × 3 × 5 = 30
2 × 2 × 2 × 7 = 56
This can be written as 23 × 7 = 56
3 × 3 × 11 = 99
This can be written as 32 × 11 = 99
First of all, explain verbally 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”
Reveal the fact 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, suffice to say it’s important.  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 like in the decomposition.
Link:
N9 Powers and roots
Every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers.

5. The prime factor decomposition
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The prime factor decomposition
When we write a number as a product of prime factors it is called the prime factor decomposition.
The prime factor decomposition of 100 is:
100 = 2 × 2 × 5 × 5
= 22 × 52
Verify that 2 × 2 × 5 × 5 = 100
There are 2 methods of finding the prime factor decomposition of a number.

6. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Factor trees 36 4 9 2 2 3 3
Explain that to write 36 as a product of 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.
Ask pupils to give examples.
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.
Emphasize that multiplications including 1, such as 1 × 36, do not help us to find the product of prime factors because the number we are trying to break down remains unchanged.
Let’s use 4 × 9 this time.
Next, we must find two numbers that multiply together to make 4.
Click to reveal two 2s.
2 is a prime number so we draw a circle around it.
Now find 2 numbers which multiply together to make 9.
Click to reveal two 3s.
3 is a prime number so draw a circle around it.
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.
Ask pupil how we can we write this using index notation (powers) before revealing this.
36 = 2 × 2 × 3 × 3
= 22 × 32

7. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Factor trees 36 3 12 4 3 2 2
Show this alternative factor tree for 36.
The prime factor decomposition is the same.
36 = 2 × 2 × 3 × 3
= 22 × 32

8. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Factor trees
2100 30 70 6 5 10 7
Again, 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. 2 3 2 5
2100 = 2 × 2 × 3 × 5 × 5 × 7
= 22 × 3 × 52 × 7

9. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Factor trees
780 78 10 2 39 5 2
Here is another example. 3 13
780 = 2 × 2 × 3 × 5 × 13
= 22 × 3 × 5 × 13

10. Dividing by prime numbers
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Dividing by prime numbers 2 96 2 3 2 48
96 = 2 × 2 × 2 × 2 × 2 × 3 2 24 2 12
= 25 × 3 2 6
Explain verbally 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. 3 3 1

11. Dividing by prime numbers
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Dividing by prime numbers 3 5 7 3
315 3
105
315 = 3 × 3 × 5 × 7 5 35
= 32 × 5 × 7 7 7
Talk through this example 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. 1

12. Dividing by prime numbers
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Dividing by prime numbers 2
702 2 3 13 3
351
702 = 2 × 3 × 3 × 3 × 13 3
117 3 39
= 2 × 33 × 13 13 13
Here is another example.
Ask pupils to find the prime factor decomposition of given numbers. 1

13. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM

14. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Common multiples
Multiples of 6
Multiples of 8 12 60 6 18 54 66
102… 8 16 24 24 32 40 48 48 56 64 72 72 80 88 96 96
104 … 30 42 78 90 84 36
Start by asking:
What is a multiple?
Let’s fill in the multiples of 8 on this straight strip.
Reveal the first number, 8, and allow pupils to call out each subsequent multiple before revealing it.
Now let’s fill in the multiples of 6.
Reveal the 6 and allow pupils to call out subsequent multiples as before.
What do we call the numbers where the two strips overlap?
They are called common multiples.
What is the lowest common multiple of 6 and 8?
The lowest common multiple is often called the LCM.

15. Multiples on a hundred grid
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Multiples on a hundred grid
Colour all of the multiples of 3 red, all the multiples of 4 blue and all the multiples of 10 yellow. Draw the pupils’ attention to the fact that the colours change when they overlap.
Ask pupils to identify all the common multiples of 3 and 4.
What do you notice? What is the lowest common multiple of 3 and 4?
Ask pupils to identify all the common multiples of 3 and 10.
What do you notice? What is the lowest common multiple of 3 and 10?
Ask pupils to identify all the common multiples of 4 and 10.
What do you notice? What is the lowest common multiple of 4 and 10?
Point out that since 4 and 10 share a common factor, 2, all of the common multiples of 4 and 10 are multiples of 20 (not multiples of 40 as many pupils will expect).
What is the lowest common factor of 3, 4 and 10?

16. The lowest common multiple
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The lowest common multiple
The lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers.
We can find this by writing down the first few multiples for both numbers until we find a number that is in both lists.
For example:
Multiples of 20 are :
20,
40,
60,
80,
100,
120, . . .
You may like to add that if the two numbers have no common factors (except 1) then the lowest common multiple of the two numbers will be the product of the two numbers.
For example, 4 and 5 have no common factors and so the lowest common multiple of 4 and 5 is 4 × 5, 20.
Pupils could also investigate this themselves.
Multiples of 25 are :
25,
50,
75,
100,
125, . . .
The LCM of 20 and 25 is
100.

17. The lowest common multiple
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The lowest common multiple
What is the lowest common multiple (LCM) of 8 and 10?
The first ten multiples of 8 are: 8,
16,
24,
32,
40,
48,
56,
64,
72,
80.
The first ten multiples of 10 are:
Again go through this method of finding the lowest common multiple.
10,
20,
30,
40,
50,
60,
70,
80,
90,
100.
The lowest common multiple (LCM) of 8 and 10 is
40.

18. The lowest common multiple
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The lowest common multiple
We use the lowest common multiple when adding and subtracting fractions.
Add together 4 9 5 12
and
The LCM of 9 and 12 is 36.
× 4
× 3
Establish verbally by asking for multiples that the lowest common multiple of 9 and 12 is 36.
Remind pupils that to add two fractions together they must have the same denominator.
The LCM is the lowest number that both 9 and 12 will divide into.
Ask pupils how many 9s ‘go into’ 36. Establish that we must multiply 9 by 4 to get 36 before revealing the first arrow.
Now, we’ve multiplied the bottom by 4 so we must multiply the top by 4.
Remember, if you multiply the top and the bottom of a fraction by the same number, you do not change its value. 16/36 is just another way of writing 4/9.
Repeat this explanation as you convert 5/12 to 15/36.
16/36 plus 15/36 equals 31/36. Can this fraction be simplified?
Establish that it cannot.
Link:
N12 Adding and subtracting fractions. + 4 9 5 12 = 16 15 31 36 + 36 = 36
× 4
× 3

19. Boardworks KS3 Maths 2009 N7 Prime factor decomposition, HCF and LCM
Common factor diagram
Select pupils to come to the board and drag and drop the numbers down the right hand side in such a way that they are in a circle containing a multiple of that number (24, 36 or 40).
If the number divides into two of these numbers then it must be placed in the area where the two circles overlap.
If the number is a factor of 24, 36 and 40, it must be placed in the region where all three circles overlap.
Numbers which do not divide into 24, 36 or 40 must be placed around the outside of the circle.
Once the diagram is complete ask questions such as:
What are the common factors of 36 and 40?
What are the common factors of 24 and 30?
What is the highest common factor of 24 and 40?”
If we used different numbers would there always be a number in the section where the three circles overlap?
Establish that 1 would always be in the central overlapping section because 1 is a factor of every whole number.

20. The highest common factor
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The highest common factor
The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists.
For example:
Factors of 36 are : 1, 2, 3, 4, 6, 9,
12,
18,
36.
Point out that 3 is a common factor of 36 and 45. As is is 1. However, 9 is the highest common factor.
Factors of 45 are : 1, 3, 5, 9,
15,
45.
The HCF of 36 and 45 is 9.

21. The highest common factor
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The highest common factor
What is the highest common factor (HCF) of 24 and 30?
The factors of 24 are: 1, 2, 3, 4, 6, 8,
12,
24.
The factors of 30 are:
Once the factors have been revealed remind pupils about factor pairs.
24 is equal to 1 × 24, 2 × 12, 3 × 8, and 4 × 6.
30 is equal to 1 × 30, 2 × 15, 3 × 10, and 5 × 6. 1, 2, 3, 5, 6,
10,
15,
30.
The highest common factor (HCF) of 24 and 30 is 6.

22. The highest common factor
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
The highest common factor
We use the highest common factor when cancelling fractions.
Cancel the fraction 36 48
The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12.
Talk through the use of the highest common factor to cancel fractions in one step.
Link:
N10 Fractions
÷12 36 48 3 = 4
÷12

23. Using prime factors to find the HCM and LCM
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Using prime factors to find the HCM and LCM
We can use the prime factor decomposition to find the HCF and LCM of larger numbers.
Find the HCF and the LCM of 60 and 294. 2 60 2
294 2 30 3
147 3 15 7 49
Recap on the method of dividing by prime numbers introduced in the previous section. 5 5 7 7 1 1
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7

24. Using prime factors to find the HCM and LCM
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Using prime factors to find the HCM and LCM
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7 60
294 2 7 2 3 5 7
We can find the HCF and LCM by using a Venn diagram.
We put the prime factors of 60 in the first circle. Any factors that are common to both 60 and 294 go into the overlapping section.
Click to demonstrate this. Point out that we can cross out the prime factors that we have included from 294 in the overlapping section to avoid adding then twice.
We put the prime factors of 294 in the second circle.
The prime factors which are common to both 60 and 294 will be in the section where the two circles overlap.
To find the highest common factor of 60 and 294 we need to multiply together the numbers in the overlapping section.
The lowest common multiple is found by multiplying together all the prime numbers in the diagram.
HCF of 60 and 294 =
2 × 3
= 6
LCM of 60 and 294 =
2 × 5 × 2 × 3 × 7 × 7 =
2940

25. Using prime factors to find the HCM and LCM
Boardworks KS3 Maths 2009
N7 Prime factor decomposition, HCF and LCM
Using prime factors to find the HCM and LCM
Use this activity to practice using a Venn diagram to find the HCF and LCM of two given numbers.