Prime Factorisation · Hook

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Ever wondered what numbers are built from?

A house in Enugu is built from blocks; every whole number bigger than 1 is built from prime numbers multiplied together. Aisha discovered that 72 is really just 2 × 2 × 2 × 3 × 3 — and that makes HCF and LCM questions much easier.

Revision notes

📒 Prime Factorisation: lesson notes

Break any number down into its prime building blocks and write the answer neatly in index form.

A house in Enugu is built from blocks; every whole number bigger than 1 is built from prime numbers multiplied together. Aisha discovered that 72 is really just 2 × 2 × 2 × 3 × 3 — and that makes HCF and LCM questions much easier.

Divide by primes until you reach 1

To find the prime factors of a number, keep dividing by the smallest prime that works: 2, then 3, then 5, then 7.

For 72: 72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1.

The primes you divided by are the building blocks: 72 = 2 × 2 × 2 × 3 × 3.

🔑 Key idea: Keep dividing by primes (2, 3, 5, 7, …) until you reach 1.

Index form is a short way to write it

Instead of writing 2 × 2 × 2, we write 23. The small number is the index (or power): it tells you how many times the prime is multiplied.

So 72 = 23 × 32. Check: 23 = 8 and 32 = 9, and 8 × 9 = 72 ✓.

Careful: 23 means 2 × 2 × 2 = 8, not 2 × 3 = 6.

🔑 Key idea: Index form: 2 × 2 × 2 × 3 × 3 = 23 × 32.

Every factor must be prime

72 = 8 × 9 is true, but it is not a prime factorisation, because 8 and 9 are not prime. Keep going until every factor is prime.

Whichever way you start, you always end up with the same primes — each number has only one prime factorisation.

🔑 Key idea: Check your answer by multiplying it out.

Worked example

Express 216 as a product of prime factors in index form.

  1. Step 1: Divide by primes

    216 ÷ 2 = 108

    108 ÷ 2 = 54

    54 ÷ 2 = 27

    27 ÷ 3 = 9

    9 ÷ 3 = 3

    3 ÷ 3 = 1

  2. Step 2: Write the primes with powers

    216 = 2 × 2 × 2 × 3 × 3 × 3

    = 23 × 33

    Why? The power tells you how many times each prime is multiplied.

Answer: 23 × 33

Summary

  • Every whole number bigger than 1 can be written as a product of primes.
  • Divide by the smallest prime again and again until you reach 1.
  • Index form groups repeated primes: 2 × 2 × 2 = 23.
  • A power means repeated multiplication, not multiplying by the index.
  • Check by multiplying the primes back together.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.