Good Learning Co.
SavannahYear 8
← Maths
MathsGCSE maths: number, accuracy and standard form30 min★★★ difficulty

Prime factorisation, HCF and LCM

Every whole number greater than one is built from primes in exactly one way. That single fact is the tool behind HCF, LCM and simplifying surds.

Lesson overview

What you'll learn in this lesson

Apply number operations, accuracy and standard form to solve GCSE-style problems.

Key learning points

  • • Writing a number as a product of primes
  • • HCF and LCM from prime factors
  • • Why examiners ask this

This lesson at a glance

  • 30 minutes
  • 15 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

numberoperationsaccuracystandardgcse-style

Scroll down — the lesson carries on below

Part 1 of 15 · Discover7%
1

Watch & discover

Part 1 of 15

Watch first

Prime factorisation, HCF and LCM

An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.

In a nutshell

Apply number operations, accuracy and standard form to solve GCSE-style problems.

2

Learning cycle

Part 2 of 15

Learning cycle 1 of 2

Part 1 · Writing a number as a product of primes

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 15

Learn

Writing a number as a product of primes

Divide repeatedly by the smallest prime that fits, then collect the factors using index notation: 360 = 2³ × 3² × 5. A factor tree and repeated division give the same result, and the answer is unique regardless of the route taken.

4

Explore the idea

Part 4 of 15

Learn

HCF and LCM from prime factors

The highest common factor is the product of the primes both numbers share, taking the lower index of each. The lowest common multiple takes every prime that appears, at the higher index. A Venn diagram makes this visible: the overlap gives HCF and the whole diagram gives LCM.

5

Quick check

Part 5 of 15

Quick check

360 as a product of primes is

6

Quick check

Part 6 of 15

Quick check

The HCF is found by taking shared primes at the

7

Reset break

Part 7 of 15

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
8

Learning cycle

Part 8 of 15

Learning cycle 2 of 2

Part 2 · Why examiners ask this

A short piece of teaching, then a check to make sure it has landed.

9

Explore the idea

Part 9 of 15

Learn

Why examiners ask this

HCF and LCM questions are usually dressed as contexts: two buses leaving together, tiles fitting a floor, ribbons cut into equal lengths. 'When do they coincide again' is an LCM question; 'largest equal size' is an HCF question. Naming which one you need is half the marks.

10

Quick check

Part 10 of 15

Quick check

Two buses leave every 12 and 18 minutes. They next leave together after

11

Quick check

Part 11 of 15

Quick check

The largest identical pieces from ribbons of 84 cm and 120 cm are

12

Explore the idea

Part 12 of 15

Worked example

Worked answer: find the HCF and LCM of 84 and 120 (4 marks)

84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5 (1). HCF takes shared primes at the lower index: 2² × 3 = 12 (1). LCM takes every prime at the higher index: 2³ × 3 × 5 × 7 (1) = 840 (1). Checking: 84 × 120 = 10,080 = 12 × 840, which confirms both answers.

13

Challenge round

Part 13 of 15

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

14

Mastery quiz

Part 14 of 15

Marked quiz

End of lesson quiz: Prime factorisation, HCF and LCM

4 questions, marked with the reasoning shown. No timer.

  1. 1. 360 as a product of primes is

  2. 2. The HCF is found by taking shared primes at the

  3. 3. Two buses leave every 12 and 18 minutes. They next leave together after

  4. 4. The largest identical pieces from ribbons of 84 cm and 120 cm are

15

Lesson round-up

Part 15 of 15

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 15 · Watch & discover

Good Learning Co. — personalised UK curriculum learning, built around what each learner loves.