## What you need to know

### Prime Factors

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, whereas both 2 and 5 are prime. Thus, 2 and 5 are the prime factors of 10.

**Note:** It is important to become extra familiar with the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, as these are likely to appear a lot and will really help with topics such as this.

Along with factor trees, it is also useful to know how basic Venn diagrams work for this topic.

### Factor Tree’s

Every positive whole number has a prime factorisation – a list of prime numbers that, when multiplied together, give you the original number.** **In the example above, 10 has only two prime factors, 2 and 5, and the resulting prime factorisation of 10 is 2\times 5. In more complicated cases, we use something called a factor tree to find a number’s unique prime factorisation. An example of which is shown below.

### Highest Common Factor – HCF

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. Factor trees can be useful in helping to determine the HCF of two numbers, especially as the numbers get larger.

### Lowest Common Multiple – LCM

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 multiple of both of these numbers is 35, therefore 35 is the LCM of 5 and 7.

Like HCF, to work out the LCM of larger numbers, it is easiest to use a Venn diagram.

### Example 1:

Determine the prime factorisation of 60.

To construct a factor tree think of 2 numbers which multiply together to make 60 – here, we’ve gone with 30 and 2. 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 as shown in the factor tree below.

### Example 2:

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.

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.

To find the HCF is to multiply the numbers in the intersection:

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

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.

### Example Questions

1) Find the unique prime factorisation of 72

Write your answer using index notation.

The prime factors of a number can be displayed using a prime factor tree.

The prime factorisation of 72 is,

72=2\times2\times2\times3\times3

Written in index notation, the answer is,

72=2^3\times3^2

2) Find the prime factorisation of 140

Write your answer using index notation.

The prime factors of a number can be displayed using a prime factor tree.

The prime factorisation of 140 is,

140=2\times2\times5\times7

Written in index notation, the answer is,

140=2^2\times5\times7

3) Find the HCF and LCM of 24 and 40

First, we have to find the prime factorisation of 24 and of 40:

Prime factors of 24: 2\times2\times2\times3

Prime factors of 40: 2\times2\times2\times5

To find the **HCF**, find any prime factors that are in **common **between both numbers.

**HCF** = 2\times2\times2=8

Next, cross any numbers used so far off from the products.

Prime factors of 24: \cancel{2}\times\cancel{2}\times\cancel{2}\times3

Prime factors of 40: \cancel{2}\times\cancel{2}\times\cancel{2}\times5

To find the **LCM**, multiply the **HCF** by all the factors that have not been crossed out so far.

**LCM** = 8\times3\times5=120

4) Find the HCF and LCM of 495 and 220

The prime factors of both 495 and 220 can be displayed using prime factor trees.

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 the factors of 495 and the other containing the factors of 220. Any prime factors shared by these two numbers are to be placed in the intersection.

495=3\times3\times\cancel5\times\cancel11

220=2\times2\times\cancel5\times\cancel11

The **HCF** can be calculated by multiplying the numbers in the intersection together,

\text{HCF }=5\times11=55

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

5) Find the HCF and LCM of 32, 152 and 600

First, we have to find the prime factorisation of 32, 152 and of 600:

Prime factors of 32=2\times2\times2\times2\times2

Prime factors of 152= 2\times2\times2\times19

Prime factors of 600= 2\times2\times2\times3\times5\times5

Then we can place each prime factor in the correct circle in the Venn diagram. Any common factors should be placed in the intersections of the circles.

The highest common factor (**HCF**) is found by **multiplying together the numbers in the intersection** of all three of the circles.

**HCF** = 2\times2\times2=8

The lowest common multiple (**LCM**) is found by **multiplying together the numbers from all sections** of the circles.

**LCM** = 2\times2\times2\times2\times2\times3\times5\times5\times19=45600

### Worksheets and Exam Questions

#### (NEW) Prime Factors, HCF, LCM - Exam Style Questions - MME

Level 4-5#### HCF LCM Product Of Primes - Drill Questions

Level 4-5#### BIDMAS and Prime Factors - Drill Questions

Level 4-5#### LCM HCF Factors and Prime Numbers - Drill Questions

Level 4-5#### Factors Multiples Primes - Drill Questions

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