1. Prime Factor Decomposition
What does this mean?
Prime
Factor
Decomposition
Prime Numbers
A number that divides exactly into another
number (no remainder),
Break down into smaller parts
So it is the process of breaking numbers down into the prime
factors that make it up.
This allows us to write a number as a Product of its Prime Factors
A ‘product’ is the answer to a multiplication so 10 is the product of
2 and 5 because 2 x 5 = 10

2. Prime Factor Decomposition
Write the number 15 as a product of its prime factors:
What are the factors of 15?
1 x 15 = 15
3 x 5 = 15
1 and 15 are not prime numbers
3 and 5 are prime numbers
So writing 15 as a product of its Prime factors means writing
3 x 5 = 15
However:
8 can be written as the product of 2 and 4 2 x 4 = 8
but 4 is not a prime number so 2 x 4 is not writing 8 as a
product of its prime factors.
4 can be written as the product of 2 and 2 2 x 2 = 4
Therefore 8 can be written as 2 x 2 x 2 = 8
This is now writing 8 as a product of its prime factors

3. Prime Factor Decomposition
How do we find all the prime factors of a number:
Start with the smallest prime factor
Find the prime factors of 24
What is the smallest prime factor of 24? 2 24
What do we multiply by 2 to get 24? 12 2 12
What is the smallest prime factor of 12? 2
What do we multiply by 2 to get 12? 6 2 6
What is the smallest prime factor of 6? 2
What do we multiply by 2 to get 6? 3 2 3
We have arrived at another prime number therefore the
factor tree is finished.
We can now write 24 as a product of its prime factors x x x
= 24
Or more simply in index notation 23 x 3 = 24

4. Prime Factor Decomposition
Write 28 as a product of its prime factors 28 2 14
Notice how the factors
Are written in order of size 2 7
2 x 2 x 7 = 28
Or more simply in index notation 22 x 7 = 28
Write 27 as a product of its prime factors 27
3 x 3 x 3 = 27 3 9
Or more simply in index notation 33 = 27 3 3

5. Prime Factor Decomposition
Now answer these:
Write the following as a product of its prime factors
Leave your answer in index form: 20 32 63 96
144
720
450
624
8820
If 1080 = 2x x 3y x 5z what are the values of x,y and z?
22 x 5 = 20
25 = 32
32 x 7 = 63
25 x 3 = 96
24 x 32 = 144
24 x 32 x 5= 720
2 x 32 x 52= 450
25 x 39 = 624
22 x 32 x 5 x 72= 8820
x = 3, y = 3, z = 1

6. Finding the LCM and HCF using Prime Factor Decomposition
We found that: x 5 = 20 and 25 = 32
The highest common factor is found by using the prime factors that
are common in both numbers 2
2 x 2 x 5 = x 2 x 2 x 2 x 2 = 32 2 2 2 x
= 4
So the HCF of 20 and 32 is 4
The Lowest Common Multiple is found by using HCF,
then using all the numbers that are different
2 x 2 x 5 = x 2 x 2 x 2 x 2 = 32 2 2 5 2 2 2 2 2
HCF x x x x x
= 160
So the LCM of 20 and 32 is 160

7. Prime Factor Decomposition
Now answer these by using Prime Factor Decomposition:
Find the HCF and LCM of these pairs of numbers
24 and 36
27 and 36
32 and 48
56 and 152
2 x 2 x 2 x 3 = 24
2 x 2 x 3 x 3 = 36
HCF = 2 x 2 x 3 = 12
LCM = 12 x 2 x 3 = 72
3 x 3 x = 27
2 x 2 x 3 x 3 = 36
HCF = 3 x = 9
LCM = 9 x 2 x 2 x 3 = 108
2 x 2 x 2 x 2 x 2 = 32
2 x 2 x 2 x 2 x 3 = 48
HCF = 2 x 2 x 2 x 2 = 16
LCM = 16 x 2 x = 96
2 x 2 x 2 x 7 = 56
2 x 2 x = 152
HCF = 2 x = 4
LCM = 4 x 7 x = 1092