KS3 · Maths

Factors and multiples

Which numbers go exactly into 54? Most people find a few and stop. By the end of this lesson, you'll have a method that never misses one.

Sort it

Factors of 36, factors of 54 — or both?

A factor of a number is a whole number that divides into it exactly, with no remainder. For each number, tick every set it belongs to. Some numbers belong to both — and one belongs to neither.

  • A Factors of 36
  • B Factors of 54
  1. 1
  2. 2
  3. 3
  4. 4
  5. 6
  6. 8
  7. 9
  8. 12
  9. 18
  10. 27
  11. 36
  12. 54

Find them all

From scratch: every factor of 54

In the Venn diagram you checked numbers someone handed you. Now find every factor of 54 on your own. Work in factor pairs: two whole numbers that multiply to make 54.

  1. Start with the pair everyone forgets: 1 × 54 = 54, so 1 and 54 are a factor pair.Picture 54 counters laid out in a rectangle. One row of 54 always works — and every factor pair is another rectangle you could make, like 2 rows of 27.
  2. missing step
Which line is step 2?

Spot the slip

When the table runs out

Use a 10 by 10 multiplication table to list all the factors of 24.

A student's answer — which line goes wrong?

Now flip it round

Where the multiples of 3 and 4 meet

Multiples are the other way to read a multiplication: the nth multiple of a number is the number × n. Drag each multiple to its place on the line. Watch for two landing on exactly the same spot.

Find the pattern

Multiples of 2, 4 and 5

Tick each box where the number is a multiple of the column's number — in other words, where it divides exactly with no remainder. Then look down the columns.

12
18
20
35
44
50

What do you really think?

Does any multiplication count?

Four students are arguing about these facts: 4 × 7.5 = 30, 2 × 7.5 = 15 and 1.5 × 10 = 15.

Which idea is closest to what you think right now?
How sure are you?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

10 × 2 = 20 says two things at once: 2 and 10 are factors of 20, and 20 is a multiple of 2 and of 10.

What you need to know

  • A factor of a number is a whole number that divides into it exactly, leaving no remainder.
  • A multiple of a number is that number multiplied by a whole number: 5, 10, 15, 20 … are multiples of 5.
  • In factor × factor = product, the product is a multiple of each factor.
  • A common factor is a factor of two or more numbers; a common multiple is a multiple of two or more numbers.

The big picture

Every multiplication of whole numbers can be read two ways: in 10 × 2 = 20, 2 and 10 are factors of 20, and 20 is a multiple of 2 and of 10. Find all the factors of a number in pairs, starting with 1 and the number and working upwards until the pairs repeat. Find common factors by comparing factor lists, and common multiples by listing multiples until they match — remembering that factors and multiples only come from multiplying whole numbers.

Key points

1Find all the factors of a number in pairs: start with 1 and the number, test 2, 3, 4 … in order, and stop when the pairs would repeat the other way round.
21 is a factor of every whole number, but 1 is only a multiple of 1.
3Every number has infinitely many multiples, so a list or a multiplication table can never show them all.
4The nth multiple of a number is the number × n. To work backwards, divide by n.
5Find common factors by listing the factors of each number and comparing; on a Venn diagram they go in the overlap.
6Find common multiples by listing the multiples of each number and picking out the numbers that appear in every list.
7Because 2 is a factor of 4, every multiple of 4 is also a multiple of 2.

Worked example

Problem

Find the common multiples of 5, 8 and 10 that are less than 100.

⚠ Watch out

Leaving out the pair 1 and the number itself when listing factors. For 36 the list of pairs must start with 1 and 36 — which is why working systematically from 1 upwards matters.

🧠

Memory hook

Factors fit INTO a number; multiples grow OUT of it. 1, 2, 3 and 6 fit into 6. 6, 12, 18 … grow out of it — and never stop.

✓

Check yourself

Without looking — find all the factor pairs of 80, and say how you know when you can stop.

Flashcards

(15)
What is a factor of a number?
A whole number that divides into it exactly, with no remainder.
What is a multiple of a number?
The number multiplied by a whole number. The multiples of 5 are 5, 10, 15, 20, …
In 10 × 2 = 20, name the factors and the multiple.
2 and 10 are factors of 20. 20 is a multiple of 2 and a multiple of 10.
Is 7.5 a factor of 15? (2 × 7.5 = 15)
No — factors must be whole numbers. The factors of 15 are 1, 3, 5 and 15.
Is 30 a multiple of 4? (4 × 7.5 = 30)
No — 7.5 isn't a whole number. 4 × 7 = 28 and 4 × 8 = 32 jump straight over 30.
Which number is a factor of every whole number?
1.
Is 1 a common multiple of all numbers?
No. 1 is only a multiple of 1.
When listing factor pairs, when can you stop?
When the next number to test is already in a pair you've found — the pairs would repeat the other way round.
What are the factor pairs of 12?
1 and 12, 2 and 6, 3 and 4. They match the arrays 1 by 12, 2 by 6 and 3 by 4 (a 4 by 3 array is the same one turned round).
How many multiples does a number have?
Infinitely many — no list or multiplication table can show them all.
How do you find the 7th multiple of a number?
Multiply the number by 7. (To go backwards from the 7th multiple, divide by 7.)
What are the common multiples of 7 and 5 below 100?
35 and 70.
What are the common factors of 20 and 30?
1, 2, 5 and 10.
Why is every multiple of 4 also a multiple of 2?
Because 2 is a factor of 4.
On a Venn diagram of the factors of two numbers, where are the common factors?
In the overlap (the intersection).

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 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.