Sheet № 09 · Foundation + Higher · AQA · Edexcel · OCR
Prime Factorisation, HCF and LCM –
Prime factorisation, HCF (highest common factor), and LCM (lowest common multiple) form a core Number topic tested on every GCSE Maths exam board at both Foundation and Higher tier. These skills are needed for simplifying fractions, solving problems involving repeated events, and working with ratios. This page explains what prime factoris
§Key definitions
Prime factorisation
means writing a number as a product of its prime factors. Every whole number greater than 1 can be expressed uniquely as a product of primes — this is known as the Fundamental Theorem of Arithmetic.
Question:
Find the HCF and LCM of 36 and 90.
Answer:
HCF = 18, LCM = 180
Q1 (Foundation):
Write 84 as a product of its prime factors. Give your answer in index notation.
Q2 (Foundation):
Find the HCF of 48 and 72.
§Formulas to memorise
For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b
Prime factorisation uses index notation: e.g. 360 = 2³ × 3² × 5
Highest Common Factor (HCF): — The largest number that divides exactly into two or more numbers. Also called the greatest common divisor.
Lowest Common Multiple (LCM): — The smallest number that is a multiple of two or more numbers.
Prime factorisation: means writing a number as a product of its prime factors. Every whole number greater than 1 can be expressed uniquely as a product of primes — this is known as the Fundamental Theorem of Arithmetic.
Worked example
Find the HCF and LCM of 36 and 90.
Working:
⚠ Common mistakes
- ✗Confusing HCF and LCM. HCF uses the lowest powers of common primes; LCM uses the highest powers of all primes. A helpful mnemonic: HCF = Highest is wrong, it is the Highest Common Factor but found using the lowest shared powers.
- ✗Not breaking a number down completely to primes. 12 = 4 × 3 is not fully factorised because 4 is not prime. You must write 12 = 2² × 3.
- ✗Forgetting to include primes that only appear in one number when finding the LCM. For example, if one number has a factor of 5 and the other does not, the 5 must still be included in the LCM.
- ✗Writing the factorisation without index notation. Examiners expect you to use powers: write 2³ × 3 × 5, not 2 × 2 × 2 × 3 × 5.
✦ Exam tips
- →Learn to draw factor trees quickly. Practise until you can factorise numbers up to 200 in under a minute.
- →Venn diagrams are very popular in exam mark schemes. If a question says "use prime factorisation to find the HCF and LCM", a Venn diagram is the clearest way to present your working.
- →Know when a problem requires HCF vs LCM. HCF questions usually involve splitting things into equal groups or finding the largest shared quantity. LCM questions involve repeated events or finding when things coincide. Our formulas guide lists the key relationships.
- →Verify your answer using HCF × LCM = product of the two numbers. For example, 18 × 180 = 3240, and 36 × 90 = 3240. They match, so the answer is correct.
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