**Prime factor decomposition** *KS3 Revision*

## What you need to know

**Things to remember:**

- Learning the prime numbers less than 30 is always helpful.
- A prime factor decomposition is all of the prime numbers multiplied together that make the number.
- If you can divide by a prime number to start with, it will make the process quicker.
- Number that end in a 0 can always be divided by 10.

A factor is a number that divides another number and gives a whole number. For example, because 32\div8=4, then 8 is a factor of 32. This means that 4 is a factor of 32 as well. We call these two numbers, 4 and 8, a factor pair. We can also think of the factor pairs as multiplying:

4\times8=32

Factors of 32:

1 2 4 8 16 32

A **prime factor decomposition** is all of the prime numbers that multiply together to make the original number. So, to find a prime factor decomposition, it is helpful to know your prime numbers.

Prime Numbers

2

3

5

7

11

13

17

19

*Find the prime factor decomposition of 420*

We can find a prime factor decomposition in two.

**Method 1:** Prime Factor Tree

We can do this in four steps:

**Step 1: **Start by writing your number at the top middle of the page.

**Step 2: **Think about two numbers that multiply together to make your number, writing these down to the sides, connecting with straight lines, and circling any prime numbers.

*Tip:**Try to divide by a prime number to start with, as this will speed up the process!*

*Tip: **If your number is even, you can divide by 2, which is prime.*

**Step 3: **Repeat **Step 2 **on the numbers that haven’t been circled until the ends of each branch are circled.

**Step 4: **Write all of the prime (circled) numbers together as a multiplication.

*Tip: **You can write this the multiplication with powers.*

2\times2\times3\times5\times7=2^2\times3\times5\times7

**Method 2: **Multiplying through.

Instead of drawing out the prime factor tree, you can write it as a multiplication, breaking up number into factors until you are left with prime numbers.

420

420=6\times70

420=2\times3\times70

420=2\times3\times5\times14

420=2\times3\times5\times2\times7

420=2\times3\times5\times2\times7=2^2\times3\times5\times7

## Example Questions

**Question 1:** Find the prime factor decomposition of 212

212=2\times2\times2\times3\times3\times3=2^3\times3^3

**Question 2:** Find the prime factor decomposition of 400

400

400=4\times100

400=2\times 2\times 100

400=2\times 2\times 10\times 10

400=2\times 2\times 2\times 5\times 10

400=2\times 2\times 2\times 5\times 2\times 5 =2^4\times5^2