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

Answer

 

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

Question 2: Find the prime factor decomposition of 400

Answer

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