KS3 · Maths

Highest common factor and lowest common multiple

What's the biggest number that divides exactly into both 240 and 200? Listing every factor would take ages. Split each number into primes, and one diagram hands you the answer.

Maths · Number

One diagram, two answers: 240 and 200

240 = 2 × 2 × 2 × 2 × 3 × 5 and 200 = 2 × 2 × 2 × 5 × 5. Pair the factors off: if BOTH numbers have it, it goes in the overlap. If only one number has it, it goes in that number's own part. Where does each factor go?

  • A Prime factors of 240
  • B Prime factors of 200
  1. First 2
  2. Second 2
  3. Third 2
  4. Fourth 2
  5. The 3
  6. First 5
  7. Second 5

Maths · Number

Factor or multiple? Think in primes

12 = 2² × 3 and 18 = 2 × 3². (The small raised number is an exponent: 2² means 2 × 2.) For each number below, tick every column that is true.

2 × 3
2² × 3
2 × 3²
2³ × 3
2² × 3²
2³ × 3²

Maths · Number

Spot the slip

Find the HCF of 840 and 2772 using a Venn diagram.

A student's answer — which line goes wrong?

Maths · Number

Now with three numbers

Find the HCF and the LCM of 84, 180 and 63.

  1. 84 = 2² × 3 × 7, 180 = 2² × 3² × 5, 63 = 3² × 7
  2. missing step
Which line is step 2?

Maths · Number

HCF or LCM: which did Maya find?

Maya's Venn diagram for 35 and 45 is correct. 35 = 5 × 7 and 45 = 3 × 3 × 5. The overlap holds 5. 35's own part holds 7. 45's own part holds 3 and 3. She writes: HCF = 7 × 5 × 3 × 3 = 315.

What do you think has happened?
How sure are you?

Maths · Number

HCF or LCM? Decide before you calculate

Does each problem need the HCF or the LCM?

Still to sort

Needs the HCF (0)

Splitting quantities into the largest possible equal groups or lengths, with none left over

Where the line is: You're breaking things DOWN into pieces, so you're looking for a factor.

Needs the LCM (0)

Building up by repeating: buying packs, or events that repeat

Where the line is: You're building UP by repeating, so you're looking for a multiple.

6 of 6 still to sort.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Break two numbers into primes, sort them into one Venn diagram, and read off both answers.

What you need to know

  • A factor divides into a number exactly; a multiple is the number times 1, 2, 3 and so on.
  • For small numbers you can list factors: 24 has 1, 2, 3, 4, 6, 8, 12, 24 and 40 has 1, 2, 4, 5, 8, 10, 20, 40, so their HCF is 8. For big numbers like 240 and 400, listing is slow and prime factors are quicker.
  • Prime factorisation writes a number as a product of primes, e.g. 240 = 2⁴ × 3 × 5.
  • In a Venn diagram of prime factors, a factor both numbers share goes in the overlap once.
  • HCF = product of the overlap. LCM = product of everything in the diagram.
  • Splitting into the largest equal groups needs the HCF; repeating until things line up needs the LCM.

The big picture

Write each number as a product of prime factors, then sort those factors into a Venn diagram: a factor both numbers have goes in the overlap once. The highest common factor (HCF) is the product of the overlap. The lowest common multiple (LCM) is the product of everything in the diagram. Splitting into the largest equal groups needs the HCF; things that repeat until they line up need the LCM.

Key points

1The highest common factor (HCF) is the biggest number that is a factor of all the numbers.
2The lowest common multiple (LCM) is the smallest number that is a multiple of all the numbers.
3An exponent shows repeated multiplication: 5² means 5 × 5.
4A common factor uses only the lower power of each prime; a common multiple must contain each number completely.
5For three numbers, the HCF is the centre of the diagram and the LCM is still everything in it.
6Check a Venn diagram by multiplying each circle back to its original number.
7An HCF can never be bigger than the numbers you started with.

Worked example

Problem

Find the HCF and the LCM of 360 and 2100.

⚠ Watch out

Mixing up the two answers. The HCF is the overlap only; the LCM is everything in the diagram. If your 'HCF' is bigger than one of your starting numbers, it can't be a factor, so you've probably worked out the LCM instead.

🧠

Memory hook

The HCF hides in the middle; the LCM takes the lot.

✓

Check yourself

45 = 3² × 5 and 60 = 2² × 3 × 5. What goes in the overlap? What are the HCF and the LCM? (Overlap: 3 and 5. HCF = 15, LCM = 180.)

Flashcards

(15)
What is a multiple of a number?
The number multiplied by 1, 2, 3 and so on. The multiples of 6 are 6, 12, 18, 24, …
What is a common factor?
A number that divides exactly into all of the numbers you are given.
What is a common multiple?
A number that is a multiple of every one of the numbers you are given.
What does HCF stand for, and what is it?
Highest common factor: the biggest number that is a factor of all the numbers.
What does LCM stand for, and what is it?
Lowest common multiple: the smallest number that is a multiple of all the numbers.
What is prime factorisation?
Writing a number as a product of prime numbers, for example 20 = 2 × 2 × 5.
In 2³, what is the small raised 3 called, and what does it mean?
The exponent. It means 2 is multiplied by itself three times: 2 × 2 × 2.
Where does a prime factor that both numbers share go in the Venn diagram?
In the overlap, written once.
How do you read the HCF from a Venn diagram of prime factors?
Multiply together the factors in the overlap.
How do you read the LCM from a Venn diagram of prime factors?
Multiply together every factor in the whole diagram.
When can a prime power be part of a common factor?
Only up to the lower power: 11² and 11⁴ share 11², but not 11⁴.
How do you check a Venn diagram of prime factors?
Multiply everything in each circle. Each circle must give back its original number.
Three numbers: where is their HCF in the Venn diagram?
In the centre, where all three circles overlap.
A problem splits things into the largest equal groups. HCF or LCM?
HCF: you're breaking things down into factors.
A problem is about things that repeat until they happen together. HCF or LCM?
LCM: you're building up in multiples.

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.

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How this lesson was checked. This KS3 Mathslesson 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 30 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.