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Prime Factorisation

Simply Maths


Prime factorisation is a way of expressing a number as a product of its prime factors. To do prime factorisation, you need to find the prime numbers that, when multiplied together, give you the original number. Here's a step-by-step guide on how to do prime factorisation:


Let's use the number 36 as an example.


Find the Smallest Prime Divisor

Start with the smallest prime number, which is 2. Check if your number is divisible by 2. If it is, divide the number by 2 and write down the factor. If it's not divisible by 2, move on to the next prime number (3), and so on.


In our example, 36 is divisible by 2, so we divide it by 2:

36 ÷ 2 = 18

So, we have the factor 2.


Continue Dividing by Prime Factors

Repeat the process with the quotient from the previous step (18). Find the smallest prime factor that divides the quotient and continue dividing until you reach a prime number.


18 is divisible by 2 again:

18 ÷ 2 = 9

So, we have another factor of 2.


Now, continue with the quotient (9):

9 is not divisible by 2 but is divisible by 3:

9 ÷ 3 = 3

So, we have another factor of 3.


Continue with the quotient (3):

3 is already a prime number, so we don't need to divide further.


Write Down the Prime Factors

Now that you have completely factored the number, write down all the prime factors you found. In this example, the prime factorisation of 36 in expanded form is:


2 × 2 × 3 × 3


In exponent form: 2² × 3²


Factor trees help to make the process and result clearer (notice I've circled the prime factors):

36 prime

Let's do another example:


Example 2: Prime Factorisation of 48

  1. Start with the number: 48.

  2. Divide by the smallest prime factor (2): 48 ÷ 2 = 24.

  3. Divide again by 2: 24 ÷ 2 = 12.

  4. Divide by 2 again: 12 ÷ 2 = 6.

  5. Divide by the next smallest prime (3): 6 ÷ 3 = 2.

  6. Finish since 2 is prime.

As a factor tree:

48 prime

Now, write down the prime factorisation:

In expanded form: 2 × 2 × 2 × 2 × 3

In exponent form: 2⁴ × 3


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