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
- 19 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 19
Visual introductionPicture this
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.
In a nutshell
Apply number operations, accuracy and standard form to solve GCSE-style problems.
Learning cycle
Part 2 of 19
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.
Explore the idea
Part 3 of 19
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.
Reset break
Part 4 of 19
Pause
That's sitting 1 of 5 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.
Explore the idea
Part 5 of 19
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.
Quick check
Part 6 of 19
Quick check
Part 7 of 19
Reset break
Part 8 of 19
Pause
That's sitting 2 of 5 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.
Learning cycle
Part 9 of 19
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.
Explore the idea
Part 10 of 19
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.
Quick check
Part 11 of 19
Reset break
Part 12 of 19
Pause
That's sitting 3 of 5 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.
Quick check
Part 13 of 19
Explore the idea
Part 14 of 19
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.
Challenge round
Part 15 of 19
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
Reset break
Part 16 of 19
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 19
Game · Recall cards
360 as a product of primes is
Card 1 of 4
Mastery quiz
Part 18 of 19
Marked quiz
End of lesson quiz: Prime factorisation, HCF and LCM
4 questions, marked with the reasoning shown. No timer.
1. 360 as a product of primes is
2. The HCF is found by taking shared primes at the
3. Two buses leave every 12 and 18 minutes. They next leave together after
4. The largest identical pieces from ribbons of 84 cm and 120 cm are
Lesson round-up
Part 19 of 19
Lesson round-up
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Ask LunaPart 1 of 19 · Watch & discover