N4NumberFoundation & HigherGrades 2–5

Factors, Multiples & Primes

Factors divide into a number exactly; multiples are the number's times table; primes have exactly two factors. Every whole number greater than 1 can be built from primes in exactly one way, and that single fact gives a fast, reliable method for the highest common factor (HCF) and lowest common multiple (LCM).

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Primes are the building blocks of multiplication, the way atoms are the building blocks of matter. Writing $360 = 2^3 \times 3^2 \times 5$ tells you everything about 360's factors at a glance.

HCF and LCM questions on AQA papers are often disguised as problems: tiles fitting exactly, buses leaving together, packs of items that must match up. Spotting "largest that divides both" (HCF) versus "first time they coincide" (LCM) is half the marks.

Factors, Multiples and PrimesCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Identify factors, multiples and prime numbers
Write a number as a product of prime factors in index form
Find the HCF and LCM of two numbers
Solve problems involving HCF and LCM

Factors, multiples and primes

A $\textbf{factor}$ of $n$ divides $n$ exactly. Find them in pairs: $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ , $6 \times 6$ .

A $\textbf{multiple}$ of $n$ is in its times table: multiples of 7 are $7, 14, 21, \ldots$

A $\textbf{prime}$ has exactly two factors, 1 and itself. The primes under 30 are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ . 1 is not prime, and 2 is the only even prime.

Example 1From the list

15, 17, 21, 24, 27, 29

, write down (a) the prime numbers (b) the multiples of 3 (c) a factor of 48.

1(a) 17 and 29 have only two factors.

2(b)

15, 21, 24, 27

are in the 3 times table.

3(c)

48 \div 24 = 2

exactly.

Answer

(a)

17, 29

(b)

15, 21, 24, 27

(c)

24

Your turnSame method, new numbers — try it before moving on.

How many factors does 36 have?

Your answer

Show hint

List them in pairs: $1 \times 36$ , $2 \times 18$ , …

Show worked solution

Solution

1Pairs: $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ , $6 \times 6$ .
2Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$ — that is 9 (6 is counted once).

Answer: $9$

Tip — Square numbers are the only numbers with an odd number of factors — one factor pairs with itself.

Prime factorisation

Use a factor tree: split the number into any factor pair, and keep splitting until every branch ends in a prime. Circle the primes.

Write the answer as a product using powers for repeated primes, in ascending order: $360 = 2^3 \times 3^2 \times 5$ .

Circled primes:

2, 2, 2, 3, 3, 5

, so

360 = 2^3 \times 3^2 \times 5

Example 1Write 84 as a product of its prime factors in index form.

84 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 7

Answer

84 = 2^2 \times 3 \times 7

Your turnSame method, new numbers — try it before moving on.

Write 150 as a product of its prime factors in index form.

Your answer

Type it like 2^2 x 3 x 7 (ascending order).

Show worked solution

Solution

1 $150 = 2 \times 75 = 2 \times 3 \times 25 = 2 \times 3 \times 5 \times 5$ .
2In index form: $2 \times 3 \times 5^2$ .

Answer: $2 \times 3 \times 5^2$

HCF and LCM

The $\textbf{HCF}$ is the largest number that divides both. The $\textbf{LCM}$ is the smallest number that both divide into.

Using prime factors: the HCF multiplies the primes the numbers $\textbf{share}$ (lowest power of each shared prime). The LCM multiplies $\textbf{every}$ prime that appears (highest power of each). A Venn diagram shows this neatly.

60 = 2^2 \times 3 \times 5

and

72 = 2^3 \times 3^2

. HCF = middle

= 2 \times 2 \times 3 = 12

. LCM = everything

= 5 \times 2 \times 2 \times 3 \times 2 \times 3 = 360

Example 1Find the HCF and LCM of 60 and 72.

60 = 2^2 \times 3 \times 5

;

72 = 2^3 \times 3^2

2HCF: shared primes, lowest powers:

2^2 \times 3 = 12

3LCM: all primes, highest powers:

2^3 \times 3^2 \times 5 = 360

Answer

HCF

12

, LCM

360

Your turnSame method, new numbers — try it before moving on.

Find the LCM of 18 and 30.

Your answer

Show worked solution

Solution

1 $18 = 2 \times 3^2$ and $30 = 2 \times 3 \times 5$ .
2LCM $= 2 \times 3^2 \times 5 = 90$ .

Answer: $90$

Example 2One bus leaves every 12 minutes and another every 20 minutes. They both leave at 9:00 am. When do they next leave together?

1"Next leave together" is a common multiple — the LCM of 12 and 20.

12 = 2^2 \times 3

20 = 2^2 \times 5

, so LCM

= 2^2 \times 3 \times 5 = 60

Answer

10:00 am

Your turnSame method, new numbers — try it before moving on.

A room measures 240 cm by 390 cm. It is covered with square tiles, the largest possible size that fit exactly with no cutting. What is the side length of a tile, in cm?

Your answer

Show hint

The largest number dividing both lengths exactly is the HCF.

Show worked solution

Solution

1 $240 = 2^4 \times 3 \times 5$ and $390 = 2 \times 3 \times 5 \times 13$ .
2HCF $= 2 \times 3 \times 5 = 30$ cm.

Answer: $30$ cm

A hidden check

For any two numbers, $\text{HCF} \times \text{LCM} =$ the product of the numbers: $12 \times 360 = 4320 = 60 \times 72$ . The shared primes are counted once in each, and every prime is counted exactly twice overall.

Think like an examiner

"Give your answer in index form" means powers are required: $2 \times 2 \times 2 \times 3$ loses a mark; $2^3 \times 3$ scores.
Show your prime factor trees or lists — method marks are available even if the final multiplication slips.
In problem questions, write whether you are finding the HCF or LCM and why.

Key ideas

360 = 2^3 \times 3^2 \times 5

Prime factorisation in index form.

\text{HCF: shared primes, lowest powers}

Largest number dividing both.

\text{LCM: all primes, highest powers}

Smallest number both divide into.

Watch out for these

Counting 1 as a prime number.

1 has only one factor, so it is not prime.

Giving the product of primes as

2 + 2 + 3 + 7

It is a product:

2^2 \times 3 \times 7

Mixing up HCF and LCM: the HCF of 12 and 20 is 60.

HCF is at most the smaller number: 4. LCM is 60.

Stopping a factor tree at a non-prime:

84 = 4 \times 21

Keep going:

2 \times 2 \times 3 \times 7

Stretch yourself

$A = 2^3 \times 3 \times 5^2$ and $B = 2 \times 3^2 \times 5$ . (a) Find the HCF of $A$ and $B$ . (b) Find the smallest whole number $k$ such that $Ak$ is a square number.

Hint — A square number has every prime raised to an even power.

Reveal solution

Solution(a) Shared primes, lowest powers:

2 \times 3 \times 5 = 30

. (b)

A = 2^3 \times 3^1 \times 5^2

. To make every power even we need one more 2 and one more 3, so

k = 6

and

6A = 2^4 \times 3^2 \times 5^2 = 3600 = 60^2

\blacksquare

Questions students ask

Is there a quick way to test if a number is prime?

Try dividing by each prime up to its square root. For 97, test 2, 3, 5 and 7 (

\sqrt{97} < 10

). None divide exactly, so 97 is prime.

Does it matter which factor pair I start my tree with?

No. Every route ends with the same primes — that is the uniqueness of prime factorisation.

Can I find the HCF by listing factors instead?

Yes, for small numbers. Prime factors are faster and more reliable for larger numbers, and are required when the numbers are given in index form.

Key takeaways

Factors divide exactly; multiples are in the times table; primes have exactly two factors.
Use a factor tree to write a number as a product of primes in index form.
HCF: multiply the shared primes (lowest powers).
LCM: multiply all the primes (highest powers).
"Largest that fits into both" → HCF; "first time together" → LCM.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 33 marks

Write 180 as a product of prime factors. Give your answer in index form.[3]
Show mark scheme
- M1A correct method, e.g. a factor tree with at least two correct splits
- A1 $2, 2, 3, 3, 5$ seen
- A1 $2^2 \times 3^2 \times 5$
Answer: $2^2 \times 3^2 \times 5$

Question 2Grade 43 marks

Burgers come in packs of 12. Buns come in packs of 18. Kai wants exactly the same number of burgers and buns.

What is the smallest number of packs of burgers he can buy?[3]
Show mark scheme
- M1Lists multiples of both or finds the LCM of 12 and 18
- A1LCM $= 36$
- A13 packs of burgers
Answer: 3 packs

Examiner note

The LCM (36) is not the final answer — the question asks for the number of packs.

Question 3Grade 52 marks

$X = 2^4 \times 3^2 \times 7$ and $Y = 2^2 \times 3^3 \times 5$ . Find the HCF of $X$ and $Y$ , and the LCM of $X$ and $Y$ . Leave your answers in index form.[2]
Show mark scheme
- B1HCF $= 2^2 \times 3^2$
- B1LCM $= 2^4 \times 3^3 \times 5 \times 7$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Which of these is a prime number?
- $31$
- $21$
- $27$
- $1$
Show answer and explanation
Answer: $31$
31 has only the factors 1 and 31.
Which of these is a factor of 42?
- $14$
- $84$
- $4$
- $8$
Show answer and explanation
Answer: $14$
$42 \div 14 = 3$ .
What is the HCF of 16 and 24?
- $8$
- $4$
- $48$
- $2$
Show answer and explanation
Answer: $8$
8 is the largest number dividing both.
What is the LCM of 6 and 8?
- $24$
- $48$
- $2$
- $14$
Show answer and explanation
Answer: $24$
$6, 12, 18, 24$ and $8, 16, 24$ .
$72$ as a product of prime factors is…
- $2^3 \times 3^2$
- $2^2 \times 3^3$
- $8 \times 9$
- $2 \times 36$
Show answer and explanation
Answer: $2^3 \times 3^2$
$8 \times 9 = 2^3 \times 3^2$ .
How many prime numbers are there between 20 and 40?
- $4$
- $3$
- $5$
- $6$
Show answer and explanation
Answer: $4$
$23, 29, 31, 37$ .
The HCF of two numbers is 6 and their LCM is 72. One number is 18. What is the other?
- $24$
- $12$
- $36$
- $4$
Show answer and explanation
Answer: $24$
HCF × LCM = product: $432 \div 18 = 24$ .
What is the smallest $k$ so that $12k$ is a square number?
- $3$
- $2$
- $12$
- $4$
Show answer and explanation
Answer: $3$
$12 = 2^2 \times 3$ needs another 3: $36 = 6^2$ .

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