Sheet № 82 · Foundation + Higher · AQA · Edexcel · OCR
HCF and LCM –
Finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) is a fundamental Number topic tested on every GCSE Maths exam board. These skills underpin work with fractions, ratios, and algebraic fractions, so mastering them is essential for both Foundation and Higher tiers.
§Key definitions
Question:
Find the HCF and LCM of 24 and 36.
Answer:
HCF = 12, LCM = 72
Q1 (Foundation):
Find the HCF of 28 and 42.
Q2 (Foundation):
Find the LCM of 6 and 10.
Q3 (Higher):
Find the HCF and LCM of 84 and 120.
§Formulas to memorise
HCF = product of shared prime factors (the intersection of the Venn diagram)
LCM = product of all prime factors in the Venn diagram (each used the maximum number of times)
Useful check: HCF × LCM = product of the two original numbers
HCF — = multiply all the numbers in the overlap.
LCM — = multiply all the numbers in the entire Venn diagram.
Worked example
Find the HCF and LCM of 24 and 36.
Working:
⚠ Common mistakes
- ✗Confusing HCF and LCM. The HCF is always smaller than or equal to both numbers. The LCM is always greater than or equal to both. If your HCF is bigger than one of the numbers, you have mixed them up.
- ✗Missing a prime factor in the decomposition. If you stop the factor tree too early (e.g. leaving a 6 instead of breaking it into 2 × 3), your HCF and LCM will be wrong.
- ✗Forgetting to use the HCF × LCM check. HCF × LCM should equal the product of the two numbers. Use this to verify your answer quickly.
✦ Exam tips
- →Draw the Venn diagram neatly in the exam — examiners often award a mark specifically for a correct Venn diagram.
- →When a question says "use prime factorisation," you must show the factor trees or repeated division. The listing method will not earn full marks.
- →Bus timetable and alarm clock problems are classic LCM contexts. Light pattern problems (two lights flashing at different intervals) also use LCM.