Prime factor decomposition Revision | KS3 Maths Resources

## 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.

$$420$$

$$420=6\times70$$

$$420=2\times3\times70$$

$$420=2\times3\times5\times14$$

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

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

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## Example Questions

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

$$400$$

$$400=4\times100$$

$$400=2\times 2\times 100$$

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

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

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

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