Mastering Prime Factorisation and Product of Primes

Understanding Prime Factorisation

In mathematics, every composite number can be broken down into a unique set of prime numbers that multiply together to create the original value. This process is known as prime factorisation. Think of prime numbers as the 'atoms' of the number system; just as atoms build molecules, prime numbers build all other integers.

Mastering this skill is vital for your KS3 maths journey. It is not just a standalone topic; it is the essential tool you will use to calculate the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers. Understanding how to express a number as a product of its primes will save you time and help you avoid errors in more complex algebraic problems later in your studies.

What are Prime Numbers?

Before we begin, we must define our building blocks. A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. It is important to remember that 1 is not a prime number, and 2 is the only even prime number.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. If you can recognise these quickly, you will find the process of factorisation much smoother.

The Factor Tree Method

A factor tree is the most reliable way to find the prime factors of any number. The goal is to keep splitting a number into factor pairs until every branch ends in a prime number.

Worked Example 1: Factorise 60

1. Start with 60. Pick any two factors, for example, 6 and 10.
2. Neither 6 nor 10 is prime, so split them further.
3. Split 6 into 2 and 3. Both are prime, so circle them.
4. Split 10 into 2 and 5. Both are prime, so circle them.
5. Collect all the circled numbers: 2, 3, 2, and 5.

We write this as a product: $2 \times 2 \times 3 \times 5$. To check, $2 \times 2 = 4$, $4 \times 3 = 12$, and $12 \times 5 = 60$. The calculation is correct.

Using Index Notation

In your exams, you will often be asked to write your answer in index form. This is a shorthand way of writing repeated multiplication. For example, instead of writing $2 \times 2 \times 2$, we write $2^3$.

Worked Example 2: Factorise 72 in Index Form

1. Start with 72. Split into 8 and 9.
2. Split 8 into 2 and 4. Split 4 into 2 and 2. (We now have three 2s).
3. Split 9 into 3 and 3. (We now have two 3s).
4. The prime factors are 2, 2, 2, 3, and 3.
5. Group them: $2 \times 2 \times 2 \times 3 \times 3$.
6. Write in index form: $2^3 \times 3^2$.

The Ladder Method

If you find factor trees messy, the ladder method (or repeated division) is a great alternative. You divide the number by the smallest possible prime number repeatedly until you reach 1.

For 72:

• $72 \div 2 = 36$
• $36 \div 2 = 18$
• $18 \div 2 = 9$
• $9 \div 3 = 3$
• $3 \div 3 = 1$

The numbers on the left side ($2, 2, 2, 3, 3$) are your prime factors.

Common Mistakes to Avoid

1. Forgetting 1 is not prime: Students often include 1 in their prime factor list. Remember, a prime number must have exactly two distinct factors.
2. Stopping too early: Ensure you continue splitting until every branch ends in a prime number. If you stop at a composite number like 4 or 6, your answer will be incomplete.
3. Calculation errors: Always perform a quick check by multiplying your final prime factors together to see if they equal the original number.
4. Ignoring index form: If a question asks for index form, ensure you use powers. Writing $2 \times 2 \times 3$ instead of $2^2 \times 3$ may lose you marks.

Frequently Asked Questions

Is 1 a prime number? No, 1 is not a prime number because it does not have two distinct factors; it only has one.

Does the order of factors matter? No, the order does not matter. $2 \times 3 \times 5$ is the same as $5 \times 3 \times 2$. However, it is standard practice to write them in ascending order.

Can I use different starting factors for a tree? Yes. Whether you start 60 as $6 \times 10$ or $2 \times 30$, you will always end up with the same prime factors ($2^2 \times 3 \times 5$).

Conclusion

Prime factorisation is a fundamental skill that unlocks your ability to solve more complex number problems with confidence. By practising factor trees and index notation, you are building a strong mathematical foundation for your GCSEs and beyond.

Ready to see these concepts in action? Head over to MathInstructor AI to generate a free, narrated animated lesson on prime factorisation and watch these numbers break down step by step.