Factors, Multiples & Primes
A factor divides exactly into a number. A multiple is the result of multiplying a number by an integer. A prime number has exactly two factors: 1 and itself. Prime factorisation means writing a number as a product of its prime factors — for example, 60 = 2² × 3 × 5. You use prime factorisation to find the HCF (highest common factor) and LCM (lowest common multiple).
Full topic guide: the detailed syllabus page with worked examples and common mistakes lives at studyvector.co.uk/gcse/maths/number/factors-multiples-primes.
Topic preview: Factors, Multiples & Primes
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Topic explanation
A factor divides exactly into a number. A multiple is the result of multiplying a number by an integer. A prime number has exactly two factors: 1 and itself. Prime factorisation means writing a number as a product of its prime factors — for example, 60 = 2² × 3 × 5. You use prime factorisation to find the HCF (highest common factor) and LCM (lowest common multiple).
Factors, Multiples & Primes is easiest to revise when it is treated as a precise exam behaviour, not a loose note-taking category. In GCSE Mathematics, the goal is to recognise how the topic appears in a question, identify the command word, and decide what evidence, method, or vocabulary earns marks. StudyVector keeps this page tied to AQA · Edexcel · OCR language where coverage is available, then routes practice towards the same topic so revision moves from explanation into retrieval.
A strong revision session starts with a short recall check. Write down the rule, definition, process, or method linked to Factors, Multiples & Primes before looking at any notes. Then answer one exam-style prompt and compare your answer with the mark-scheme logic: did you make a clear point, support it with the right step, and avoid drifting into a nearby topic? This matters because many lost marks come from almost-correct answers that do not match the expected structure.
Use this guide as the first layer: understand the topic, look at the worked examples, complete the mini quiz, then move into full practice. The full StudyVector practice loop is designed to capture whether mistakes are caused by knowledge, method, language, or timing. That distinction is important. If the error is factual, you need reteaching. If the error is method-based, you need a worked retry. If the error is wording, you need command-word calibration. That is how Factors, Multiples & Primes becomes a controlled revision target rather than another page in a folder.
Lost marks → repair task
Why marks are usually lost here
These are the error patterns StudyVector looks for after an attempt. The goal is not a generic explanation; it is one repair move and one follow-up question.
Unit, formula, or method slip
Examiner move: Select the correct method and keep units, substitutions, signs, and rounding visible.
Repair drill: Redo the calculation or method line slowly, naming the formula before substituting values.
Missing chain of reasoning
Examiner move: Show the link between point, method, evidence, and conclusion instead of jumping to the final line.
Repair drill: Write the missing because/therefore step, then retry one isomorphic question.
Timing breakdown
Examiner move: Match answer length to marks and avoid over-writing low-mark questions.
Repair drill: Set a one-mark-per-minute cap and write a compact version before expanding.
Mini quiz
Use these checks before full practice. They test topic recognition, exam technique, and whether you can connect the explanation to a marked response.
1. What should you check first when a Factors, Multiples & Primes question appears in GCSE Mathematics?
- A.The command word and the exact topic focus
- B.The longest paragraph in your notes
- C.A memorised answer from a different topic
2. Which revision action gives the strongest evidence that Factors, Multiples & Primes is improving?
- A.Rereading the explanation twice
- B.Answering a timed exam-style question and reviewing lost marks
- C.Highlighting every key phrase in the topic notes
Sample questions
Topic-specific public question previews are still being reviewed. We keep them off public pages until the topic match is safe.
Exam tips
- Read the command word carefully — "explain" needs reasons; "state" expects a short fact.
- For Factors, Multiples & Primes, show structured working even when you are practising multiple choice — it builds accuracy under time pressure.
- Mark yourself against the mark scheme style: one clear point per mark, in logical order.
- Come back to this topic after a day or two; short spaced reviews beat one long cram.
Worked examples
Example 1
Modelled exam response
Find the HCF and LCM of 36 and 90. Prime factorise: 36 = 2² × 3², 90 = 2 × 3² × 5. HCF = 2¹ × 3² = 18 (lowest powers of shared primes). LCM = 2² × 3² × 5 = 180 (highest powers of all primes).
Example 2
Identify the task before answering
Question type: a Factors, Multiples & Primes prompt asks for a clear response in GCSE Mathematics. Step 1: underline the command word. Step 2: name the exact part of Factors, Multiples & Primes being tested. Step 3: decide whether the mark scheme wants a definition, method, explanation, comparison, or calculation. Why it works: most weak answers fail before the content starts because they answer the topic generally rather than the exact exam task.
Example 3
Turn feedback into a repair task
Suppose your answer shows partial understanding but loses marks for precision. First, rewrite the missing mark as a short target: "I need to state the mechanism, unit, reason, or evidence explicitly." Then answer one similar question without notes. Finally, compare the second attempt with the first and check whether the same mark was recovered. Why it works: Factors, Multiples & Primes improves faster when feedback creates a specific retry, not another passive reading session.
Next revision routes from this subject
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Common mistakes
- Forgetting that 1 is NOT a prime number — it only has one factor.
- Missing a branch in the factor tree and getting an incomplete prime factorisation.
- Confusing HCF and LCM — HCF uses the lowest powers of shared primes, LCM uses the highest powers of all primes.
- Not writing the final answer in index form when the question asks for it.
FAQs
How do I find HCF and LCM using prime factorisation?
Write each number as a product of prime factors. For HCF, take the lowest power of each shared prime. For LCM, take the highest power of every prime that appears in either factorisation.
Is 1 a prime number?
No. A prime number must have exactly two distinct factors. The number 1 only has one factor (itself), so it is not prime.
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