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

Factors and Multiples is a core Number subtopic in Cambridge IGCSE Mathematics (0580/0607), and it appears both as standalone questions and inside ratio, fraction and scheduling problems. If you can list factors quickly, distinguish HCF from LCM, and use prime factorisation, you avoid the slips that cost two or three marks per paper. This guide explains exactly what the subtopic covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• A factor divides exactly into a number; a multiple is what you get when you multiply by a whole number.
• HCF is the largest shared factor; LCM is the smallest shared multiple.
• Prime factorisation is the safest method for HCF and LCM with larger numbers.
• Signal words matter: “largest identical groups” → HCF; “next time together” → LCM.

What are Factors and Multiples in Cambridge IGCSE Maths?

Factors and Multiples is the study of how whole numbers divide into one another and how they build up through multiplication. In Cambridge IGCSE Mathematics it covers listing factors, finding common factors and multiples, highest common factor (HCF) and lowest common multiple (LCM), often via prime factorisation. It is tested with short calculations and practical word problems.

You can read the full explanation, worked examples and notes on Tutopiya’s Factors and Multiples 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
Factor	Divides exactly into a number (no remainder)	“List all the factors of 48”
Multiple	Number × integer	”Write down the first four multiples of 7”
Common factor	A factor shared by two or more numbers	”Find the common factors of 12 and 18”
HCF	Highest common factor	”Find the HCF of 36 and 84”
LCM	Lowest common multiple	”Work out the LCM of 6, 8 and 12”

How to find the HCF and LCM — step by step

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

1. Write each number as a product of prime factors. Example: 36 = 2² × 3², and 84 = 2² × 3 × 7.
2. For the HCF, take the lowest power of each shared prime. Shared: 2² and 3 → HCF = 2² × 3 = 12.
3. For the LCM, take the highest power of every prime that appears. 2² × 3² × 7 = 252.
4. Sanity-check: HCF ≤ smaller number; LCM ≥ larger number.

Once you have worked through a few, test yourself with the free Factors and Multiples 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 into the largest equal groups	”largest”, “greatest”, “maximum number of identical…”
LCM	Events that repeat and next coincide	”least”, “smallest”, “together again”, “at the same time”
Factors only	Listing divisors	”List all the factors of…”
Multiples only	Times-table style	”Write down the first… multiples of…”

Factors and Multiples in past-paper wording: command words that matter

Most lost marks come from misreading the command word. Cambridge reuses the same phrasing, so decoding the wording is half the battle.

Command word / phrase	What the question wants	Typical stem
List all the factors of	Every divisor, usually in pairs	”List all the factors of 60.”
Write down the first … multiples of	Start of the times table	”Write down the first five multiples of 9.”
Find the HCF / highest common factor	Largest shared divisor	”Find the highest common factor of 24 and 60.”
Find the LCM / lowest common multiple	Smallest shared multiple	”Find the lowest common multiple of 8 and 12.”
Work out / Calculate	Produce a value, showing method	”Work out the LCM of 6, 9 and 15.”
Show that	Prove a given result with working	”Show that the HCF of 18 and 30 is 6.”

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

Practising the wording — not just the maths — is what full marks reward.

1. “List all the factors of 60.” Work in pairs: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 → {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}. Reward: complete list, no duplicates.
2. “Buses leave a station every 15 minutes and every 20 minutes. They leave together at 09:00. Work out the next time they leave together.” “Next … together” → LCM. 15 = 3×5, 20 = 2²×5 → LCM = 60 minutes → 10:00. Reward: method then correct time.
3. “Find the highest common factor (HCF) of 48 and 72.” 48 = 2⁴×3, 72 = 2³×3² → lowest shared powers: 2³×3 = 24. Reward: prime factors or factor pairs, then answer.

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

How Factors and Multiples connects to the rest of Number

Prime factorisation in this subtopic is the same skill taught in Number Theory. HCF feeds into simplifying fractions in Fractions, Decimals and Percentages. 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) with multiples (multiply by).
• Finding HCF when the question asks for LCM because they ignored signal words.
• Missing factors when listing — always work in factor pairs.
• Using a factor tree but forgetting index form when the question demands it.

When you need more support

If HCF/LCM word problems keep tripping you up, work through the Number Theory 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 Factors and Multiples hard in Cambridge IGCSE Maths? No — the ideas are simple. The challenge is choosing HCF or LCM correctly and using prime factorisation for larger numbers.

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

What is the difference between a factor and a multiple? A factor divides exactly into a number (3 is a factor of 12). A multiple is the result of multiplying (12 is a multiple of 3).

How do I revise Factors and Multiples effectively? Read the subtopic notes, work a few HCF/LCM word problems by hand, then take the Factors and Multiples quiz to check your method.

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