Product of Prime Factors
Master product of prime factors for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers writing a number as a product of its prime factors and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Every integer has a unique prime factorisation — use a factor tree.
Free downloads
These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Topic overview
Every whole number above \(1\) can be written as a product of prime numbers, and that expression is unique. This is the fundamental theorem of arithmetic, and it is the reason prime factorisation is so useful.
The standard technique is a factor tree. Split the number into any factor pair, then keep splitting until every branch ends in a prime. It does not matter which pair you start with — the final set of primes is always the same.
The answer is normally written using index notation, so \(360\) becomes \(2^3 \times 3^2 \times 5\). This compact form is what you then use to find highest common factors and lowest common multiples.
Revision notes
Building a factor tree
Write the number at the top and split it into any two factors. Circle a branch when it reaches a prime and keep splitting the others.
For \(60\): split into \(6 \times 10\), then \(6\) into \(2 \times 3\) and \(10\) into \(2 \times 5\). The primes at the ends are \(2, 3, 2, 5\).
Writing the answer in index form
Collect the primes, count how many times each appears, and write them in ascending order with indices.
So \(60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\). Index form is expected unless the question says otherwise.
Checking the factorisation
Multiply your primes back together and confirm you get the original number.
For \(2^2 \times 3 \times 5\): \(4 \times 3 \times 5 = 60\), which matches. This catches any branch you forgot to split.
Key points
- Every number above 1 is a unique product of primes.
- Split into factor pairs until every branch is prime.
- The starting factor pair does not affect the final answer.
- Write the answer in index form, in ascending order.
- Multiply back to check.
- The prime factorisation is used to find the HCF and LCM.
Worked examples
Example 1
Write \(84\) as a product of its prime factors.
Working
Example 2
Write \(360\) as a product of its prime factors.
Working
Example 3
Check that \(2^2 \times 3 \times 7 = 84\).
Working
Common mistakes
Leaving a composite number in the answer.
84 = 4 × 21 is not finished, because neither 4 nor 21 is prime. Keep splitting.
Writing the answer without indices.
2 × 2 × 2 × 3 × 3 × 5 should be given as 2³ × 3² × 5 when index form is asked for.
Including 1 in the factorisation.
1 is not prime and adds nothing, so it never appears.
Assuming a different starting pair gives a different answer.
The prime factorisation is unique, so any correct tree reaches the same set of primes.
Exam tips
- Circle each prime as you reach it so you know which branches are finished.
- Write the primes in ascending order in your final answer.
- Always multiply back to check.
- Keep the factorisation — you will need it for HCF and LCM questions.
Key terms
- Prime factor
- A factor that is also a prime number.
- Factor tree
- A diagram showing a number split repeatedly into factors.
- Index form
- Writing repeated factors using powers, such as \(2^3\).
- Product
- The result of multiplying numbers together.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.