GCSE · Maths · AQA · Spec 8300

Prime numbers, factors and multiples

Sort the primes inside 60 and 72 into a Venn diagram, and one picture shows the biggest number dividing both and the smallest number both divide into.

Prime recipes

What do 60 and 72 have in common?

Every whole number bigger than 1 can be made by multiplying primes together, like a recipe. Here are two: 60 = 2 × 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3. Each prime below is one ingredient. Tick A if 60's recipe uses it, B if 72's recipe does, and both if the two recipes share it. Watch the repeats: count how many 2s and how many 3s each number really has.

  • A 60
  • B 72
  1. The first 2
  2. The first 3
  3. The 5
  4. The second 2
  5. The third 2
  6. The second 3

The ingredients

Prime, or just pretending?

A prime number has exactly two factors: 1 and itself. Sort each number. A few of these are very good at looking prime.

Still to sort

Prime (0)

Exactly two factors: 1 and itself.

Where the line is: Find even one factor besides 1 and the number itself, and it isn't prime.

Not prime (composite) (0)

More than two factors.

Where the line is: Being odd doesn't make a number prime. Try dividing by 3, 7 and other primes before you decide.

Neither (0)

Fewer than two factors.

Where the line is: 1 has only one factor, itself. That's too few to be prime, and it isn't composite either.

8 of 8 still to sort.

Watch out: Odd doesn't mean prime. 9, 21, 51 and 91 are all odd, and not one of them is prime.

Your turn

Write the recipe for 504

Write 504 as a product of prime factors, in index form.

  1. 504 is even, so start with the smallest prime, 2: 504 ÷ 2 = 252.Every division has to be by a prime. Starting small keeps it tidy.
  2. missing step
Which line is step 2?

Predict, then check

Two people, one number, two different starting points.

Amir starts splitting 120 as 10 × 12. Beth starts with 2 × 60. They both keep splitting until every number is prime. What happens?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Break a whole number into primes and it only ever comes out one way. Once you have those recipes, HCF and LCM are just a matter of reading a diagram.

What you need to know

  • A prime number has exactly two factors: 1 and itself. 1 is not prime (it has only one factor), and 2 is the only even prime.
  • A factor of a number divides into it exactly: the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is in its times table: 12, 24, 36 and so on.
  • A common factor divides into both numbers, and the highest common factor (HCF) is the biggest one. A common multiple is in both times tables, and the lowest common multiple (LCM) is the smallest one.
  • Prime factor decomposition writes a number as a product of primes. Repeats are written as powers, which is called index form: 45 = 3 × 3 × 5 = 3² × 5.
  • Every whole number greater than 1 has exactly one prime factorisation, apart from order. This is the unique factorisation theorem.

The big picture

A prime has exactly two factors, 1 and itself, so 1 isn't prime and 2 is the only even prime. A factor divides into a number exactly; a multiple is in its times table. Every whole number greater than 1 is a product of primes in exactly one way, apart from order, written in index form like 504 = 2³ × 3² × 7. Put the primes two numbers share in the middle of a Venn diagram: multiply the middle for the HCF, and multiply everything for the LCM.

Key points

1To break a number into primes, keep dividing by primes (or splitting into factor pairs) until only primes are left, then write the repeats as powers.
2Put the primes two numbers share in the middle of a Venn diagram, and each number's leftover primes in its own region.
3HCF = the product of the primes in the middle. LCM = the product of every prime in the diagram. In index form, that's each shared prime at its smaller power for the HCF, and every prime at its bigger power for the LCM.
4Sense check: the HCF is never bigger than the smaller number, and the LCM is never smaller than the larger number.

Worked example

Problem

Find the HCF and the LCM of 84 and 90, reading them straight from index form without drawing a diagram.

⚠ Watch out

Swapping the HCF and the LCM. The HCF uses only the middle of the Venn diagram; the LCM uses everything. If your 'HCF' of 60 and 72 comes out as 360, it can't be right: a number bigger than 60 can't divide into 60.

🧠

Memory hook

HCF: multiply the middle. LCM: multiply the lot.

✓

Check yourself

Write 150 and 105 as products of prime factors in index form. Then use a Venn diagram to find their HCF and their LCM.

Flashcards

(6)
What makes a number prime? Is 1 prime?
A prime has exactly two factors: 1 and itself. 13 is prime (factors 1 and 13). 1 is not prime: it has only one factor.
What's the difference between a factor and a multiple of 8?
A factor divides into 8 exactly: 1, 2, 4 and 8. A multiple is in the 8 times table: 8, 16, 24 and so on.
HCF: what is it, and how do you read it off a prime-factor Venn diagram?
The highest common factor: the biggest number that divides into both. Multiply the primes in the overlap. For 12 = 2² × 3 and 18 = 2 × 3², it's 2 × 3 = 6.
LCM: what is it, and how do you read it off a prime-factor Venn diagram?
The lowest common multiple: the smallest number in both times tables. Multiply every prime in the diagram. For 12 = 2² × 3 and 18 = 2 × 3², it's 2 × 3 × 2 × 3 = 36.
Write 2 × 2 × 2 × 5 × 5 in index form.
2³ × 5². The small raised number counts how many times that prime appears. (It equals 200.)
What does the unique factorisation theorem say?
Every whole number greater than 1 has exactly one prime factorisation, apart from order. However you start splitting a number, you end with the same primes.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More AQA GCSE Maths topics

How this lesson was checked. This AQA GCSE Maths (specification 8300)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 29 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.