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Prime Factor Decomposition - GCSE Maths Definition

Reviewed by: Jamie Wood

Last updated

5 March 2025

What is prime factor decomposition?

In GCSE maths, the process of writing a number as a product of its prime factors is known as prime factor decomposition. For example, the prime factor decomposition of 30 would be 2×3×5.

Some numbers may have a prime factor which is repeated in its product of prime factors. For example the prime factor decomposition of 40 is 2×2×2×5. This can be written more concisely using powers as 2³×5.

How do I find the prime factor decomposition of a number?

You can use a "factor tree" to find prime factors. This works by writing the number at the top and splitting this up into any two factors (they do not need to be prime), and writing them down at the end of the "branches".

Repeat this process for the numbers at the end of each branch, until you reach a prime number. Remember that a prime number only has two factors; 1 and itself. It is helpful to circle prime factors when they appear in the tree.

An example factor tree is shown below for 360. It can be seen that the prime factor decomposition for 360 is 2×2×2×3×3×5 or 2³×3²×5.

360 can be split up into 2 and 180. 180 can be split into 60 and 3. 60 splits into 20 and 3. 20 splits into 10 and 2. 10 splits into 2 and 5. — **Factor tree for 360**

A prime factor decomposition for a number is unique. For example, the only way to write 360 in prime factor decomposition form is as the product of three 2's, two 3's and one 5.