Breaking Numbers Down: The Power of Prime Factorisation

Every whole number greater than 1 has a secret identity — it can be built by multiplying prime numbers together. Think of prime numbers as the 'atoms' of maths: the smallest building blocks that cannot be split any further. Before the main lesson, warm up your brain with this question.

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and 13. A composite number is any whole number greater than 1 that is not prime — it has more than two factors. For example, 12 is composite because it can be divided by 1, 2, 3, 4, 6, and 12. Prime factorisation means writing a number as a product (multiplication) of prime numbers only. Every composite number has exactly one prime factorisation — this is called the Fundamental Theorem of Arithmetic. The easiest way to find prime factors is to draw a factor tree. Start with your number. Split it into any two factors. If a factor is prime, circle it — you are done with that branch. If a factor is composite, split it again. Keep going until every branch ends in a prime number. Index notation is a short way to write repeated multiplication. Instead of writing 2 × 2 × 2, you write 2³. The small raised number is called an exponent (or index), and it tells you how many times the base number is multiplied by itself. Worked Example — Prime factorisation of 36: Step 1: Split 36 → 6 × 6 Step 2: Split each 6 → 2 × 3 and 2 × 3 Step 3: All branches end in primes: 2, 3, 2, 3 Step 4: List them in order: 2 × 2 × 3 × 3 Step 5: Use index notation: 2² × 3² So the prime factorisation of 36 is 2² × 3². No matter which factor pairs you start with (e.g. 4 × 9 or 2 × 18), you will always reach the same answer.