Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Number Theory — primes, factors, multiples, HCF and LCM — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise Number Theory in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Number Theory revision-guide angle, while Tutopiya’s Number Theory subtopic page owns the learning resource and the free Number Theory quiz owns the practice.

Number Theory is the first subtopic in the Number unit of Cambridge IGCSE Mathematics (0580/0607), and it quietly underpins a large share of the paper. If you are confident with primes, factors, multiples, HCF and LCM, you move faster and make fewer slips everywhere else — from simplifying fractions to working with indices. This guide explains exactly what Number Theory covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• Number Theory is the study of whole numbers and their building blocks: primes, factors and multiples.
• The two highest-value skills are finding the HCF and the LCM, and the safest method for both is prime factorisation.
• HCF answers “the largest number that divides into…”; LCM answers “the next time things line up…”.
• Master this subtopic first — it makes fractions, ratio and indices noticeably easier.

What is Number Theory in Cambridge IGCSE Maths?

Number Theory is the study of whole numbers and how they are built from prime numbers — the indivisible building blocks of every integer. In Cambridge IGCSE Mathematics it covers primes, factors, multiples, highest common factor (HCF) and lowest common multiple (LCM). It is foundational rather than advanced, but examiners test it precisely.

You can read the full explanation, worked examples and notes on Tutopiya’s Number Theory subtopic page before you attempt questions.

The core ideas you must master

These five ideas appear again and again. Learn what each one means and the exam phrasing that signals it.

Idea	What it means	How the exam uses it
Prime number	A number with exactly two factors: 1 and itself (2, 3, 5, 7, 11…)	”Write 84 as a product of its prime factors”
Factor	A number that divides exactly into another	”List all the factors of 36”
Multiple	The result of multiplying a number by an integer	”Find the first three common multiples of 6 and 8”
HCF	Highest common factor of two or more numbers	”Find the HCF of 24 and 60”
LCM	Lowest common multiple of two or more numbers	”Two bells ring together every…”

How to find the HCF and LCM — step by step

The most reliable method for both HCF and LCM is prime factorisation, because it works even with large numbers where listing factors is slow.

1. Write each number as a product of prime factors (use a factor tree). Example: 24 = 2³ × 3, and 60 = 2² × 3 × 5.
2. For the HCF, take the lowest power of each shared prime. Shared primes are 2 and 3 → 2² × 3 = 12.
3. For the LCM, take the highest power of every prime that appears. 2³ × 3 × 5 = 120.
4. Sanity-check: the HCF should be ≤ the smaller number, and the LCM should be ≥ the larger number.

Once you have worked through a few, test yourself with the free Number Theory quiz — it tells you fast whether the method has actually stuck.

HCF vs LCM: which one does the question want?

Students lose marks by computing the right value for the wrong quantity. Use the signal words in the question to decide.

You want the…	When the question is about…	Typical signal words
HCF	Splitting or grouping into the largest equal parts	”largest”, “greatest”, “maximum number of identical…”
LCM	Events that repeat and next coincide	”least”, “smallest”, “ring/start together again”, “at the same time”

Number Theory in past-paper wording: command words that matter

Most lost marks in Number Theory come from misreading the command word — the instruction that tells you exactly what to do. Cambridge and Edexcel reuse the same phrasing across papers, so learning to decode the wording is half the battle. These are the command words you will see on Number Theory questions and what each one demands.

Command word / phrase	What the question wants	Typical Number Theory stem
Write … as a product of its prime factors	Full prime factorisation, usually in index form	”Write 360 as a product of its prime factors.”
Express … in the form …	Give the answer in a specified layout (e.g. 2ⁿ × 3)	“Express 200 in the form 2ᵃ × 5ᵇ.”
Find the HCF / highest common factor	The largest number dividing both	”Find the highest common factor (HCF) of 24 and 60.”
Find the LCM / lowest common multiple	The smallest shared multiple	”Find the lowest common multiple (LCM) of 8 and 12.”
Show that	Prove a given result with working — the answer is given, so the method earns the marks	”Show that 84 = 2² × 3 × 7.”
Work out / Calculate	Produce a value, showing method	”Work out the LCM of 6, 9 and 15.”
Write down	State the answer; no working expected (usually 1 mark)	“Write down a prime number between 30 and 40.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the maths — is what §-level marks reward. Here is how three real-style stems are answered.

1. “Write 360 as a product of its prime factors. Give your answer in index form.” Build a factor tree → 360 = 2³ × 3² × 5. Mark-scheme reward: one mark for correct primes, one for correct index form. Writing 2 × 2 × 2 × 3 × 3 × 5 often loses the “index form” mark.
2. “Two lighthouses flash together. One flashes every 12 seconds, the other every 18 seconds. Work out after how many seconds they next flash together.” The phrase “next … together” signals LCM. 12 = 2² × 3, 18 = 2 × 3² → LCM = 2² × 3² = 36 seconds. Reward: method (prime factors or listing) then the correct value.
3. “Show that the HCF of 24 and 60 is 12.” “Show that” means the answer is given — you earn marks for method: 24 = 2³ × 3, 60 = 2² × 3 × 5, take lowest shared powers 2² × 3 = 12. Stating “12” with no working scores nothing here.

When you can recognise the wording instantly, work the full set on the Number topical past-paper questions and the Number Theory quiz to lock the method in.

How Number Theory connects to the rest of Number

Number Theory is the gateway to the wider Number unit. Prime factorisation feeds directly into the Factors and Multiples subtopic, and writing primes in index form (2³, not 2 × 2 × 2) is the same skill you will use in Exponents and Surds. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move straight from a weak subtopic into the next.

Common mistakes students make

• Confusing factors (divide into a number) with multiples (times-table of a number).
• Forgetting that 1 is not prime and that 2 is the only even prime.
• Mixing up HCF and LCM because they ignored the signal words above.
• Listing factors by hand for large numbers instead of using prime factorisation.

When you need more support

If Number Theory questions keep tripping you up — especially HCF/LCM word problems — work through the Factors and Multiples quiz and the Number topical past-paper questions to pinpoint the exact gap, then get focused help from a Cambridge IGCSE Maths tutor to fix it quickly.

Frequently asked questions

Is Number Theory hard in Cambridge IGCSE Maths? No — it is one of the more approachable subtopics. The concepts are simple, but you need to be precise with definitions (primes, factors, multiples) and choose the right method for HCF and LCM.

What is the easiest way to find the HCF and LCM? Write each number as a product of prime factors. Take the lowest shared powers for the HCF and the highest powers of all primes for the LCM. It is faster and safer than listing factors.

Is 1 a prime number? No. A prime number has exactly two different factors — 1 and itself. Because 1 has only a single factor, it is not prime. The smallest prime number is 2.

How do I revise Number Theory effectively? Read the subtopic notes, work a few examples by hand, then take the Number Theory quiz to check your method. Revisit any HCF/LCM word problems you got wrong before moving on.

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