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Mr Barton’s Maths Notes
Number
2. Prime Factors, HCF & LCM
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2. Prime Factors, HCF & LCM
1. Prime Factors
Any positive integer can be written as a product of its prime factors.
Now, that may sound complicated, but all it means is that you can break up any number
into a multiplication of prime numbers, and it’s really easy to do with Factor Trees!
Don’t Forget: 1 is NOT a prime number, so will NEVER be in your factor tree
e.g. Express 60 as a product of its prime factors
You can break the
number up however you
like:
60
x
6
60
6 x 10 or 12 x 5
10
x
12
Continue breaking up
each new number into a
multiplication
3
x
2
2
3 x 2 x 2 x 5 = 60
x
5
3
Stop when you reach a
Prime Number and put a
ring around it
Check your answer by
multiplying all the
numbers together
x
2
4
x
2
3 x 2 x 2 x 5 = 60
5
Look: Even though we started a different way, we still ended up with the same answer!
Now, it looks good if you write your answer starting with the smallest numbers:
So: 60 = 2 x 2 x 3 x 5
And if you want to be really posh, you can use indices:
So: 60 = 22 x 3 x 5
Now we’ll do a harder one, but the technique is just the same.
e.g. Express 360 as a product of its prime factors
You can break the number up
however you like. I just went for 36
x 10 because it was easy to spot
360
x
36
x
6
3
x
2
6
3
x
2
2
3 x 2 x 3 x 2 x 2 x 5 = 360
360 = 2 x 2 x 2 x 3 x 3 x 5
360 = 23 x 32 x 5
Continue breaking up each new
number into a multiplication
10
x
5
Stop when you reach a Prime
Number and put a ring around it
Check your answer by multiplying all
the numbers together
Write the numbers in order
If you can, use indices
2. Highest Common Factor
The Highest Common Factor (HCF) of two numbers, is the highest number that divides
exactly into both
3. Lowest Common Multiple
The Lowest Common Multiple (LCM) of two numbers, is the lowest number that is in the
times table of both your numbers
Now, you can find both of these by trial and error, but I will show you a better way!
e.g. Find the LCM and HCF of 24 and 40
First, use Factor Trees to express your numbers as products of their prime factors:
24
x
12
x
6
3
x
2
2
40
2
24 = 2 x 2 x 2 x 3
x
4
2
x
40 = 2 x 2 x 2 x 5
2
10
2
x
5
Now, write your answers on top of each other, like this:
24 = 2 x 2 x 2 x 3
40 = 2 x 2 x 2 x 5
Draw two inter-locking circles, and label one 24 and the other 40
24
40
2
3
2
2
5
Any numbers that appear in
both answers go in the
middle (the three 2s).
The numbers left over go in
the circle they belong to
Now, here is the clever bit:
To get the Highest Common Factor you just multiply all the numbers in the middle
So, HCF = 2 x 2 x 2 = 8
To get the Lowest Common Multiple you just multiply every number you can see
So, LCM = 3 x 2 x 2 x 2 x 5 = 120
Good luck with
your revision!