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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn8300Papers 1–3 · Higher tierNumberTypes of number, factors and multiples

Number · Subtopic 1 of 5

Types of Number, Factors and Multiples

Every whole number is built from primes, like a molecule from atoms. Writing numbers as products of primes makes highest common factors and lowest common multiples quick. Along the way you meet the families of numbers: integers, squares, cubes, rational and irrational numbers.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Recognise natural numbers, integers, primes, squares, cubes, triangle numbers, rational and irrational numbers.
Find reciprocals, square roots and cube roots.
Write a number as a product of prime factors in index form.
Find the HCF and LCM and use them in problems.

Pause and think

Two buses leave the station together at $9{:}00$ . One returns every $12$ minutes, the other every $15$ minutes. When do they next leave together?

Section 2 of 14

Core idea

Start hereWhen will the buses meet again?

Two buses leave the station together at 9:00. One returns every 12 minutes, the other every 18. When are they next both there? You could list times, but breaking 12 and 18 into their prime building blocks answers it instantly, and the same trick answers many sharing and timing puzzles.

The big idea

Every whole number above 1 is prime or a product of primes in exactly one way. HCF: each prime the fewest times it appears; LCM: the most times.

1. Kinds of number

Natural: $1,2,3,\ldots$ ; integers: also $0$ and negatives.
Rational: can be written $\frac ab$ (terminating and recurring decimals).
Irrational: cannot, e.g. $\sqrt2$ , $\pi$ .
Prime: exactly two factors, 1 and itself.
Square $1,4,9,\ldots$ ; cube $1,8,27,\ldots$ .

Edge caseIs 1 prime? No: it has only one factor. (If 1 were prime, factorisations wouldn't be unique:

6=2\times3=1\times2\times3

2. Prime factorisation

A factor tree for 360

Keep splitting until every branch ends in a prime (shown in colour). Collect them:

360=2^3\times3^2\times5

Worked example360 as a product of primes

Write 360 as a product of prime factors.

Goal: Index form.
Strategy: Divide by the smallest prime repeatedly.

$360\to180\to90\to45$
Three divisions by 2.
$45\to15\to5$
Two divisions by 3.
$360=2^3\times3^2\times5$
5 is prime: stop.

Answer

2^3\times3^2\times5

✓ Check

8\times9\times5=360

✓.

3. HCF and LCM

\text{HCF}=\prod p^{\min},\qquad\text{LCM}=\prod p^{\max},\qquad\text{HCF}\times\text{LCM}=a\times b

$p$: each prime appearing.
min/max: smaller / larger of its two powers.

Worked example360 and 84

$84=2^2\times3\times7$ . Find the HCF and LCM of 360 and 84.

Goal: HCF, LCM.
Strategy: Compare powers prime by prime.

$2:\ \min(3,2)=2,\ \max=3$
Powers of 2.
$3:\ \min(2,1)=1,\ \max=2;\ 5,7:\ \min 0,\ \max1$
Other primes.
$\text{HCF}=2^2\times3=12,\ \text{LCM}=2^3\times3^2\times5\times7=2520$
Multiply.

AnswerHCF

12

, LCM

2520

✓ Check

12\times2520=30240=360\times84

✓.

Worked exampleThe buses

Every 12 and 18 minutes, together at 9:00. Next together?

Goal: Time.
Strategy: LCM.

$12=2^2\cdot3,\quad18=2\cdot3^2$
Factorise.
$\text{LCM}=2^2\cdot3^2=36$
Highest powers.

Answer9:36.

✓ Check

36=3\times12=2\times18

✓.

Common misconceptionHCF or LCM? "Next time together", "smallest number both divide into": LCM. "Largest equal groups", "biggest tile that fits": HCF.

Pause and think

Is $\sqrt{49}$ rational?

Summary

Prime factorise by repeated division.
HCF: lowest powers; LCM: highest powers.

Prime: Exactly two factors.
HCF / LCM: Highest common factor / lowest common multiple.

HistoryEuclid's infinite primes. Around

300

BC, Euclid proved there are infinitely many primes. Suppose there were only finitely many: multiply them all together and add

1

. The new number leaves remainder

1

when divided by every prime on the list, so it must have a prime factor that isn't on the list. Contradiction!

Fun factA 41-million-digit prime. The largest known prime when this lesson was written (found in October 2024) is

2^{136\,279\,841}-1

, with over

41

million digits. It was found by a volunteer computing project that searches for primes of this special "Mersenne" form.

Section 3 of 14

Key formulae

HCF and LCM from primes

\text{HCF: shared primes, lowest powers}\qquad\text{LCM: all primes, highest powers}

A useful check

\text{HCF}\times\text{LCM}=a\times b

TrickThe factor tree. Split the number into any two factors, then keep splitting until every branch ends in a prime. Circle the primes and write them in order with powers. Different trees always give the same primes.

Section 4 of 14

Worked examples

Example 1 · HCF and LCM0 / 2 steps

Find the HCF and LCM of

36

and

60

Example 2 · Product of primes0 / 2 steps

Write

360

as a product of prime factors in index form.

Section 5 of 14

Explore the graph

When do two cycles line up?

Loading the interactive graph…

Try this

Each wave crosses the axis at every multiple of its number (with $a=12$ : at $12, 24, 36, \ldots$ ).
Where do both waves cross the axis at the same point (apart from $0$ )? The first such point is the LCM.
Try $a=4$ , $b=6$ . Is the LCM $24$ or $12$ ?

Common multiples are where both patterns repeat together.

In the real worldPrimes keep your bank details safe. Online banking uses RSA encryption, which relies on a simple fact: multiplying two huge primes is easy, but splitting their product back into those primes is extraordinarily hard.

Section 6 of 14

Common mistakes

Incorrect

1

is a prime number

Correct

1

is not prime

A prime has exactly two factors.

IncorrectHCF with the highest powers

CorrectLowest powers

The HCF must divide both numbers.

IncorrectLCM

=a\times b

always

CorrectOnly if they share no factors

12\times18=216

but the LCM is

36

Incorrect

0.333\ldots

is irrational

CorrectIt is

\tfrac13

, so rational

Recurring decimals are rational.

$1$ is a prime number

Prime numbers

you need exactly two factors to join

1

I have one. I'm sitting this out.

Section 7 of 14

Exam tip

Exam tipIndex form for products of primes. "Write

360

as a product of prime factors" expects

2^3\times3^2\times5

, not a list or a factor tree. Use multiplication signs and index notation for full marks.

Section 8 of 14

Guided practice

SA-4024-NU-G013 marksL1 FoundationNon-calculatorHCF & LCM

84=2^2\times3\times7

and

90=2\times3^2\times5

. Find the highest common factor (HCF) and the lowest common multiple (LCM) of

84

and

90

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-4024-NU1-0101 markL1 FoundationNon-calculatorTypes of number

From the list

-3,\ 0,\ 2.5,\ 7,\ \sqrt2

, how many numbers are integers?

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0071 markL1 FoundationCalculatorTypes of number

Find the cube root of

343

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0131 markL1 FoundationNon-calculatorTypes of number

Write down the reciprocal of

0.25

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0112 marksL1 FoundationNon-calculatorHCF & LCM

360=2^a\times3^b\times5

. Find

a

and

b

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0012 marksL1 FoundationCalculatorHCF & LCM

Find the highest common factor of

36

and

60

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0122 marksL2 StandardNon-calculatorHCF & LCM

Find the highest common factor of

48

and

180

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0022 marksL1 FoundationCalculatorHCF & LCM

Find the lowest common multiple of

12

and

18

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU1-0032 marksL2 StandardCalculatorHCF & LCM

Two buses leave a station together at

09{:}00

. One returns every

12

minutes and the other every

15

minutes. After how many minutes do they next leave together?

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-4024-NU1-0092 marksL2 StandardCalculatorHCF & LCM

Find the smallest positive whole number that is divisible by

4

6

and

10

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Know the number families: integers, squares, cubes, rational, irrational.
Primes have exactly two factors; $1$ is not prime.
Every number is a unique product of primes.
HCF: shared primes, lowest powers. LCM: all primes, highest powers.

Section 12 of 14

Quick quiz

Q1 · SA-4024-NU-Q011 markL1 FoundationCalculatorTypes of number

Which of these numbers is irrational?

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-4024-NU1-0061 markL1 FoundationCalculatorTypes of number

Which of these numbers is prime?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: GCSE Mathematics 8300
Where: Number
Expected: Prime factorisation, HCF and LCM, including from Venn diagrams of prime factors.
Calculator: Paper 1 non-calculator; Papers 2 and 3 calculator.
Typical style: Problem-solving questions often combine topics.
Open in: 8300 · Papers 1–3 8300 · Papers 1–3

Section 14 of 14

Past paper questions

9 questions (21 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

4024/12 · Oct/Nov 2025Question 42 marks
A multiple of 7 that rounds to 30.
Question paper, page 4 Mark scheme
4024/22 · Oct/Nov 2025Question 144 marks
LCM, HCF and cube numbers to find unknowns.
Question paper, page 9 Mark scheme
4024/12 · May/June 2025Question 14 marks
Arithmetic with negative numbers, powers and roots.
Question paper, page 3 Mark scheme
4024/12 · May/June 2025Question 103 marks
Write a number as a product of primes.
Question paper, page 8 Mark scheme
4024/22 · May/June 2025Question 23 marks
Write a number in figures, spot primes and find a reciprocal.
Question paper, page 3 Mark scheme
4024/12 · Oct/Nov 2024Question 21 mark
Temperature change with negative numbers.
Question paper, page 2 Mark scheme

Show 3 more

4024/12 · Oct/Nov 2024Question 121 mark
Rational and irrational numbers.
Question paper, page 7 Mark scheme
4024/12 · Oct/Nov 2024Question 141 mark
Prime factorisation and lowest common multiple.
Question paper, page 8 Mark scheme
4024/12 · May/June 2024Question 62 marks
Prime numbers and multiples.
Question paper, page 6 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).