Year 9 maths lesson · Factors, Multiples, Primes

Prime Factorisation

Break any whole number into its unique product of prime factors. Learn to build factor trees, express results in index form, and use prime factorisations to find all factors of a number systematically.

  • About 3–6 minutes
  • 4 quick questions as you go
  • Then about 6 minutes of practice
Practice questions → Set for Class

Worked examples

Prime factorisation of 30

  1. 30 = 6 × 5
  2. 6 isn’t prime — it’s 3 × 2
  3. 3, 2, and 5 are all prime — done
  4. 30 = 2 × 3 × 5 (primes in ascending order)

Prime factorisation of 90

  1. 90 = 9 × 10
  2. 9 = 3 × 3, and 10 = 5 × 2
  3. 90 = 2 × 3 × 3 × 5
  4. Two 3s → use index form: 90 = 2 × 3² × 5

Finding all factors of 30 from its prime factorisation

  1. 30 = 2 × 3 × 5
  2. Group differently: 2 × 15, 6 × 5, 10 × 3
  3. Plus 1 and 30 itself
  4. All 8 factors: 1, 2, 3, 5, 6, 10, 15, 30

Finding all factors of 42 from its prime factorisation

  1. 42 = 2 × 3 × 7
  2. Group differently: 2 × 21, 6 × 7, 14 × 3
  3. Plus 1 and 42 itself
  4. All 8 factors: 1, 2, 3, 6, 7, 14, 21, 42

Before this lesson

It helps to be comfortable with:

Lesson transcript

Everything the lesson says, word for word

You can break matter down into smaller and smaller parts, until you reach atoms — the smallest indivisible pieces. The atoms determine the properties — whether it’s a solid or a gas, a metal or not. You can do the same thing with whole numbers. Take forty. Forty is four times ten. Four is two times two. Ten is five times two. Two and five can’t be broken down any further — they’re prime. Just as atoms determine the properties of matter, the primes determine everything about a number — whether it’s even or odd, whether it’s a perfect square, how many factors it has.

Let’s break thirty into its prime factors. Thirty is six times five. Six isn’t prime — it’s three times two. Three is prime, two is prime, five is prime — we’re done. Thirty equals two times three times five. We write the primes in ascending order.

Now let’s try ninety. Ninety is nine times ten. Nine is three times three. And ten is five times two. So ninety is two times three times three times five. Since we have two threes, we write three squared — using index form. The prime factorisation of ninety is two times three squared times five.

Once we know the prime factorisation, we can find every factor. Take thirty equals two times three times five. Group the primes different ways. Two times the rest: three times five is fifteen. So two and fifteen are factors. Two times three is six, leaving five. Six and five are factors. Two times five is ten, leaving three. Ten and three are factors. Plus one and thirty itself. That’s eight factors altogether.

Let’s try another one. Forty-two equals two times three times seven. Group the primes. Two times the rest: three times seven is twenty-one. So two and twenty-one are factors. Two times three is six, leaving seven. Six and seven. Two times seven is fourteen, leaving three. Fourteen and three. Plus one and forty-two. Eight factors again.

Here’s a nice pattern. If a number is the product of exactly two distinct primes — p times q — then its only factors are one, p, q, and p times q. Four factors, always.

More Factors, Multiples, Primes lessons

  • Factors, Multiples and PrimesIdentify factors and multiples of a number and determine whether a number is prime.
  • HCF and LCMUse prime factorisations and Venn diagrams to find the highest common factor and lowest common multiple.

Next lesson: HCF and LCM →All Year 9 Number lessonsAll lessons