## What you need to know

Firstly, we define a **factor** of a given number to be anything that that number is divisible by. For example, 1, 2, 3, and 6 are all the factors of 6.

Furthermore, we define a **prime number** as any number greater than 1 that has only two factors: itself and 1. So, 5 is a prime number, since its only factors are 1 and 5. However, the number 6, as mentioned, is divisible by 2 and 3 as well as 1 and 6, so it is not prime. We call the number 6 a **composite number**.

Note: it would be an impossible task to learn all the prime numbers off by heart – there are an infinity of them – so it’s important to understand what they are so to be able to spot one. That said, it is useful to become extra familiar with the first few, say: 2, 3, 5, 7, 11, 13, 17, as these are likely to appear a lot.

It follows that we define a **prime factor **of any given number to be any factor that number has that is also a prime number. Consider the number 10, for example. 10 is divisible by 1, 2, 5, and 10. Clearly 10 is not prime (it is divisible by 5 and 2), and 1 is also not prime (only numbers greater than 1 can be prime), whereas both 2 and 5 are prime. Thus, 2 and 5 are the prime factors of 10.

We can extend this idea further to talk about **prime factorisations**. Every positive whole number has a prime factorisation – a list of prime numbers that, when multiplied together, give you the original number. Furthermore, this list is unique for every number! Hence, they’re often called **unique prime factorisations. **In the example above, 10 has only two prime factors, 2 and 5, and the resulting prime factorisation of 10 is 2\times 5. This is a nice example, but it gets less obvious when the numbers get bigger and are divisible by the same prime factor multiple times. In more complicated cases, we use something called a **factor tree **to find a number’s unique prime factorisation.

**Example: **Determine the prime factorisation of 60.

To construct a factor tree, write the number you’re trying to factorise, 60, somewhere on your page that has plenty of space underneath. Then, think of 2 numbers which multiply together to make 60 – here, we’ve gone with 30 and 2 (if this is impossible, then your first number must be prime). Next, draw two branches coming down off 60, and at the end of the branches write the two factors that you chose. If a factor is prime, then circle it. If a factor is not prime, then repeat the process we just did for 60 with that next number.

Looking at the factor tree on the right, you can see that we split 30 up into factors 3 and 10, circled 3 since it was prime, and proceed to split 10 up into 5 and 2, which are both prime. Once all the numbers at the ends of your branches have been circled, you are done.

**HCF & LCM**

The **highest common factor**, or **HCF**, of two numbers is the biggest number that goes into both of them. Consider the numbers 12 and 20. The factors of 12 are: 1, 2, 3, 4, 6, and 12. The factors of 20 are: 1, 2, 4, 5, 10, and 20. They have a few factors in common, but the biggest factor they have in common is 4, therefore 4 is the HCF of 12 and 20,

A **multiple** of some number is the result of multiplying it by some other number. For example, 18 is a multiple of 9, since it is the result of multiply 9 by 2.

The **lowest common multiple**, or **LCM**, of two numbers is the smallest number that is a multiple of both of them. Consider the numbers 5 and 7. Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, … and multiples of 7 are: 7, 14, 21, 28, 35, 42, … and so on. So, we can see that the first occurrence of a number which is a multiplier of both of these numbers is 35, therefore 35 is the LCM of 5 and 7.

The HCF and LCM are straightforward enough for smaller numbers – when both lists of factors and multiples are not too hard to find – but when we get to trying to find the LCM and HCF of numbers in the hundreds and thousands, we have to take a different approach.

**Example: **Find the LCM and HCF of 420 and 132.

To do this method, we require the full prime factorisations of both 420 and 132. So, we’re going to use the factor tree method we saw earlier.

As you can see on the right, 420 splits into the factors 10 and 42. 10 then splits in 2 and 5 which are both prime, whilst 42 splits into 6 and 7, of which 7 is prime. Finally, the 6 splits in 3 and 2, so the full prime factorisation of 420 is

2\times2\times3\times 5\times 7.

Going through the same process, we get that the full prime factorisation of 132 is

2\times2\times 3\times 11.

So, now that we have the prime factorisations, we need to draw a Venn diagram where one circle is for prime factors of 132 and one circle is for prime factors 420. Looking at the lists of factors, if one is shared by both numbers, then we will put it in the intersection and cross it off both lists. Then, any factors that aren’t shared, and so haven’t been crossed out, will be put in their respective circles.

Now that we’ve done this, all we need to find the HCF is to multiply the numbers in the intersection:

\text{HCF }=2\times2\times3=12.

And to find the LCM, all we need to do is multiply all the numbers now in the Venn diagram together:

\text{LCM}=5\times7\times2\times 2\times3\times11=4,620.

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### Example Questions

1) Find the unique prime factorisation of 72. Write your answer using index notation.

We will use a factor tree to determine the prime factors of 72. It’s okay if your factor tree is not identical to this – there are multiple correct ones – but you must have the same result, the same prime factorisation. So, our factor tree looks like

Therefore, the prime factorisation of 72 is

72=2\times2\times2\times3\times3

Written in index notation, the answer is

72=2^3\times3^2

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2) Find the LCM and HCF of 495 and 220.

Firstly, we will have to determine the prime factorisation of both 495 and 220, which we will do with a pair of factor trees. If your factor trees aren’t identical to these, that doesn’t mean they are wrong – there are multiple correct factor trees. As long as you get the prime factorisations right, that’s what matters.

So, the factorisation of 220 is

220=2\times2\times5\times11,

and the factorisation of 495 is

495=3\times3\times5\times11.

Now, we will draw a Venn diagram with one circle containing factors of 495 and the other containing factors of 220. We will look for any prime factors shared by these two numbers, and then we will write each shared factor into the intersection of the Venn diagram one a time, crossing it off __both__ lists of factors. Once this is done, we then place any remaining factors in their relevant circles. This looks like

Then, we find the HCF by multiplying the numbers in the intersection together:

\text{HCF }=5\times11=55.

And finally, we find the LCM by multiplying all the numbers in the Venn diagram together:

\text{LCM }=3\times3\times5\times11\times2\times2=1,980.

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