Ever wondered how to find what numbers have in common...
Maths82 views·Updated 22 Aug 2026·3 pagesUnderstanding HCF and LCM: Step-by-Step Guide
Jess Davies@jess_notes21
1 / 3
1
of 3
Understanding Prime Factors and Factor Trees
Prime numbers are the building blocks of all numbers - they only divide by themselves and 1 (like 2, 3, 5, 7, 11). Think of them as mathematical atoms that can't be broken down further.
A prime factor tree helps you break any number down into these basic building blocks. For example, to find the prime factors of 72, you keep dividing until you're left with only prime numbers.
Starting with 72, you might divide by 8 and 9, then break those down further until you get 72 = 3 × 3 × 2 × 2 × 2. In index form, this becomes 3² × 2³, which is much tidier to write.
Quick Tip: Always start your factor tree with any factors you can spot easily - there's no single "correct" way to draw it, as long as you end up with prime numbers at the bottom!
2
of 3
Finding HCF and LCM Using Prime Factors
The highest common factor (HCF) is the biggest number that divides into both your numbers exactly. The lowest common multiple (LCM) is the smallest number that both your original numbers divide into.
Let's work through finding the HCF and LCM of 36 and 56. First, break both numbers into their prime factors using factor trees: 36 = 3 × 3 × 2 × 2 and 56 = 7 × 2 × 2 × 2.
Now comes the clever bit - draw a Venn diagram with two overlapping circles. Put the prime factors that appear in both numbers (the common ones) in the overlapping section.
Remember: The Venn diagram method makes it impossible to get HCF and LCM mixed up - you'll never confuse which calculation to use!
3
of 3
Calculating Your Final Answers
For 36 and 56, the common prime factors are 2 × 2, so these go in the overlapping section of your Venn diagram. The remaining factors (3 × 3 for 36, and 7 × 2 for 56) go in the outer parts of each circle.
The HCF is simply the product of everything in the overlap: 2 × 2 = 4. This means 4 is the largest number that divides exactly into both 36 and 56.
The LCM uses every prime factor exactly once, so multiply everything across both circles: 3 × 3 × 2 × 2 × 2 × 7 = 504. This means 504 is the smallest number that both 36 and 56 divide into exactly.
Pro Tip: You can check your LCM by verifying that both original numbers divide into it exactly - 504 ÷ 36 = 14 and 504 ÷ 56 = 9, both with no remainders!
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content: Least Common Multiple (lcm)
4MathsHCF & LCM Explained
Learn how to calculate the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using Venn diagrams. This summary covers key concepts, step-by-step examples, and practical applications for mastering HCF and LCM calculations.
111,11373
MathsCalculating LCM & HCF
Explore the methods for finding the Lowest Common Multiple (LCM) and Highest Common Factor (HCF) with practical examples. This summary covers step-by-step calculations for numbers like 3, 5, 60, and 96, providing clear insights into the concepts of LCM and HCF. Ideal for students preparing for exams or needing a quick reference.
117998
Mathsgcse maths cheat sheets
basic facts and rules for gcse maths
102689
MathsGCSE Maths Revision Units 9.1 & 9.2
Comprehensive revision notes covering key concepts from GCSE Maths Units 9.1 and 9.2, including multi-step equations, LCM, HCF, significant figures, algebraic manipulation, and more. Perfect for Year 9 students preparing for exams.
91,05924
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1080,2436,325
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
102,65761
MathsYear 8 Maths AQA Exam
Explore the AQA Year 8 Term 3 Main Paper 2, featuring comprehensive solutions to key mathematical concepts including geometry, percentages, sequences, and data representation. This resource covers essential topics such as area calculations, properties of shapes, and survey analysis, making it ideal for exam preparation and revision.
83,562138
MathsTrigonometric Functions Overview
Explore the fundamentals of trigonometry, including the tangent, sine, and cosine functions. This summary covers key concepts such as SOH CAH TOA, trigonometric ratios, and methods for finding angles and sides in right triangles. Ideal for students preparing for exams or needing a quick reference.
91,77159
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
1222,0611,821
MathsGCSE Maths 2018 Exam Insights
Explore the key concepts from the 2018 GCSE Maths Paper 2, including compound interest, probability, standard form, and geometric transformations. This comprehensive summary covers essential topics such as interest rates, area calculations, and Venn diagrams, providing students with a clear understanding of the exam's requirements. Ideal for exam preparation and practice.
98,187411
MathsUnderstanding Surds
Explore the concept of surds, including their definition, examples, and methods for simplifying them. This summary covers key techniques for simplifying surds, such as identifying square factors and combining terms. Ideal for students looking to master radical expressions and enhance their understanding of square roots.
984620
MathsGCSE Maths Foundation Checklist
Comprehensive revision checklist covering essential topics for the GCSE Maths Foundation tier, including statistics, geometry, algebra, probability, and trigonometry. Perfect for students aiming to pass their exams with confidence.
113,86452
MathsFoundation Maths Exam Solutions
Explore detailed solutions for the Foundation Tier Non-Calculator Maths exam. This resource covers key concepts such as probability, volume calculations, data representation, and more. Perfect for students preparing for their GCSE Maths exam, with step-by-step explanations and examples.
1021,3461,242
Most popular content
9
SociologyComprehensive Crime & Deviance Overview
Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.
1251,7221,403
CriminologyCriminology: Crime & Punishment Overview
Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.
1254,9211,060
SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1273,9202,306
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,187125
SociologySociological Theories Overview
Comprehensive revision of key sociological theories including Functionalism, Marxism, Feminism, and Interpretivism. Explore concepts like value freedom, identity formation, and the critique of social control. Ideal for AQA A-Level Sociology students preparing for exams. This summary covers essential theories and their implications in sociology, providing a clear understanding of each perspective.
1231,546847
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12103,0863,042
SociologyMedia Studies: Key Concepts & Theories
Dive into the essential concepts and theories of media studies for AQA A-level Sociology. This comprehensive revision guide covers topics such as media influence, representations, globalization, and sociological perspectives, ensuring you grasp the critical elements needed for your exams. Perfect for students seeking to enhance their understanding of media's role in society.
1222,756515
SociologyCrime and Deviance AQA A-level sociology
AQA A-level crime and deviance topic notes
1288819
BiologyA-Level Biology Year 1 Overview
Comprehensive summary of AQA A-Level Biology Year 1, covering key topics such as cellular structure, protein synthesis, immune response, gas exchange, and more. Ideal for exam preparation and understanding biological concepts. Includes detailed insights into cellular processes, biological classification, and the circulatory system.
1215,055699
Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user
Maths82 views·Updated 22 Aug 2026·3 pagesUnderstanding HCF and LCM: Step-by-Step Guide
Jess Davies@jess_notes21
Ever wondered how to find what numbers have in common or their smallest shared multiple? Understanding HCF and LCM is crucial for solving fraction problems, simplifying expressions, and tackling exam questions with confidence.
1
of 3
Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Understanding Prime Factors and Factor Trees
Prime numbers are the building blocks of all numbers - they only divide by themselves and 1 (like 2, 3, 5, 7, 11). Think of them as mathematical atoms that can't be broken down further.
A prime factor tree helps you break any number down into these basic building blocks. For example, to find the prime factors of 72, you keep dividing until you're left with only prime numbers.
Starting with 72, you might divide by 8 and 9, then break those down further until you get 72 = 3 × 3 × 2 × 2 × 2. In index form, this becomes 3² × 2³, which is much tidier to write.
Quick Tip: Always start your factor tree with any factors you can spot easily - there's no single "correct" way to draw it, as long as you end up with prime numbers at the bottom!
2
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding HCF and LCM Using Prime Factors
The highest common factor (HCF) is the biggest number that divides into both your numbers exactly. The lowest common multiple (LCM) is the smallest number that both your original numbers divide into.
Let's work through finding the HCF and LCM of 36 and 56. First, break both numbers into their prime factors using factor trees: 36 = 3 × 3 × 2 × 2 and 56 = 7 × 2 × 2 × 2.
Now comes the clever bit - draw a Venn diagram with two overlapping circles. Put the prime factors that appear in both numbers (the common ones) in the overlapping section.
Remember: The Venn diagram method makes it impossible to get HCF and LCM mixed up - you'll never confuse which calculation to use!
3
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Calculating Your Final Answers
For 36 and 56, the common prime factors are 2 × 2, so these go in the overlapping section of your Venn diagram. The remaining factors (3 × 3 for 36, and 7 × 2 for 56) go in the outer parts of each circle.
The HCF is simply the product of everything in the overlap: 2 × 2 = 4. This means 4 is the largest number that divides exactly into both 36 and 56.
The LCM uses every prime factor exactly once, so multiply everything across both circles: 3 × 3 × 2 × 2 × 2 × 7 = 504. This means 504 is the smallest number that both 36 and 56 divide into exactly.
Pro Tip: You can check your LCM by verifying that both original numbers divide into it exactly - 504 ÷ 36 = 14 and 504 ÷ 56 = 9, both with no remainders!
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content: Least Common Multiple (lcm)
4MathsHCF & LCM Explained
Learn how to calculate the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using Venn diagrams. This summary covers key concepts, step-by-step examples, and practical applications for mastering HCF and LCM calculations.
111,11373
MathsCalculating LCM & HCF
Explore the methods for finding the Lowest Common Multiple (LCM) and Highest Common Factor (HCF) with practical examples. This summary covers step-by-step calculations for numbers like 3, 5, 60, and 96, providing clear insights into the concepts of LCM and HCF. Ideal for students preparing for exams or needing a quick reference.
117998
Mathsgcse maths cheat sheets
basic facts and rules for gcse maths
102689
MathsGCSE Maths Revision Units 9.1 & 9.2
Comprehensive revision notes covering key concepts from GCSE Maths Units 9.1 and 9.2, including multi-step equations, LCM, HCF, significant figures, algebraic manipulation, and more. Perfect for Year 9 students preparing for exams.
91,05924
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1080,2436,325
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
102,65761
MathsYear 8 Maths AQA Exam
Explore the AQA Year 8 Term 3 Main Paper 2, featuring comprehensive solutions to key mathematical concepts including geometry, percentages, sequences, and data representation. This resource covers essential topics such as area calculations, properties of shapes, and survey analysis, making it ideal for exam preparation and revision.
83,562138
MathsTrigonometric Functions Overview
Explore the fundamentals of trigonometry, including the tangent, sine, and cosine functions. This summary covers key concepts such as SOH CAH TOA, trigonometric ratios, and methods for finding angles and sides in right triangles. Ideal for students preparing for exams or needing a quick reference.
91,77159
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
1222,0611,821
MathsGCSE Maths 2018 Exam Insights
Explore the key concepts from the 2018 GCSE Maths Paper 2, including compound interest, probability, standard form, and geometric transformations. This comprehensive summary covers essential topics such as interest rates, area calculations, and Venn diagrams, providing students with a clear understanding of the exam's requirements. Ideal for exam preparation and practice.
98,187411
MathsUnderstanding Surds
Explore the concept of surds, including their definition, examples, and methods for simplifying them. This summary covers key techniques for simplifying surds, such as identifying square factors and combining terms. Ideal for students looking to master radical expressions and enhance their understanding of square roots.
984620
MathsGCSE Maths Foundation Checklist
Comprehensive revision checklist covering essential topics for the GCSE Maths Foundation tier, including statistics, geometry, algebra, probability, and trigonometry. Perfect for students aiming to pass their exams with confidence.
113,86452
MathsFoundation Maths Exam Solutions
Explore detailed solutions for the Foundation Tier Non-Calculator Maths exam. This resource covers key concepts such as probability, volume calculations, data representation, and more. Perfect for students preparing for their GCSE Maths exam, with step-by-step explanations and examples.
1021,3461,242
Most popular content
9SociologyComprehensive Crime & Deviance Overview
Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.
1251,7221,403
CriminologyCriminology: Crime & Punishment Overview
Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.
1254,9211,060
SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1273,9202,306
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,187125
SociologySociological Theories Overview
Comprehensive revision of key sociological theories including Functionalism, Marxism, Feminism, and Interpretivism. Explore concepts like value freedom, identity formation, and the critique of social control. Ideal for AQA A-Level Sociology students preparing for exams. This summary covers essential theories and their implications in sociology, providing a clear understanding of each perspective.
1231,546847
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12103,0863,042
SociologyMedia Studies: Key Concepts & Theories
Dive into the essential concepts and theories of media studies for AQA A-level Sociology. This comprehensive revision guide covers topics such as media influence, representations, globalization, and sociological perspectives, ensuring you grasp the critical elements needed for your exams. Perfect for students seeking to enhance their understanding of media's role in society.
1222,756515
SociologyCrime and Deviance AQA A-level sociology
AQA A-level crime and deviance topic notes
1288819
BiologyA-Level Biology Year 1 Overview
Comprehensive summary of AQA A-Level Biology Year 1, covering key topics such as cellular structure, protein synthesis, immune response, gas exchange, and more. Ideal for exam preparation and understanding biological concepts. Includes detailed insights into cellular processes, biological classification, and the circulatory system.
1215,055699
Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user