HCF and LCM (Edexcel GCSE Maths): Revision Notes
HCF and LCM
What are HCF and LCM?
Understanding the fundamental concepts of HCF and LCM is essential for solving many mathematical problems involving factors and multiples.
Highest Common Factor (HCF) is the largest number that divides exactly into two or more numbers without leaving a remainder.
Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.
Finding HCF and LCM using prime factors
The most reliable method for finding HCF and LCM involves breaking numbers down into their prime factors. This systematic approach works for any combination of numbers and provides a clear visual representation of the relationship between them.
Method 1: Prime factor trees
To find prime factors, create a factor tree by repeatedly dividing by prime numbers until you reach all prime factors. This method ensures you don't miss any factors and provides a clear working trail.
Worked Example: Prime Factorization
Express 108 as a product of prime factors:
Step 1: Start with the number and divide by the smallest prime
Step 2: Continue dividing by primes
Step 3: Express as a product Therefore:
Method 2: Using Venn diagrams
Once you have the prime factors of both numbers, you can use a Venn diagram to find HCF and LCM visually. This method is particularly powerful because it shows exactly which factors are common and which are unique to each number.
Worked Example: Finding HCF and LCM using Venn Diagrams
Find HCF and LCM of 108 and 240
Step 1: Find prime factors
Step 2: Draw a Venn diagram with:
- Left circle: factors of 108 only ()
- Intersection: common factors ()
- Right circle: factors of 240 only ()
Step 3: Calculate HCF
- HCF = product of all prime factors in the intersection
- Common factors:
- HCF =
Step 4: Calculate LCM
- LCM = product of all prime factors in the entire diagram
- All factors:
- LCM =
Exam tips and techniques
Important exam guidance
Proper presentation of your work is crucial for gaining full marks in examinations. Following the correct conventions will ensure your answers are clear and complete.
When writing numbers as products of prime factors:
- Always use × signs in your final answer
- Never use + signs
- Don't write just a list of prime factors
- Use index notation (powers) to simplify your answer
Common exam mistakes
Students often struggle with questions involving finding possible values. Understanding the underlying concept is essential for success in these challenging problems.
Common Pitfall: Finding Possible Values
Remember that any number containing the required prime factors exactly once will work.
Example: If 10 is the HCF of 20 and n, find possible values for n.
- Any number containing prime factors 2 and 5 exactly once will work
- Possible answers: 10, 30, 50, 70, etc.
Step-by-step process
Finding HCF
The process for finding HCF involves identifying what the numbers have in common:
- Write each number as a product of prime factors
- Identify the common prime factors
- Multiply the common factors together (using the lowest power of each)
Finding LCM
The process for finding LCM requires considering all possible factors:
- Write each number as a product of prime factors
- List all prime factors that appear in either number
- For each prime factor, use the highest power that appears
- Multiply all these together
Practice approach
When tackling HCF and LCM problems, following a systematic approach will help you avoid common mistakes and ensure accurate results:
- Start by finding prime factors using factor trees
- Use Venn diagrams to visualise the relationship
- Check your answers make sense (HCF should be smaller than both original numbers, LCM should be larger)
- Always express your final answer using index notation where possible
Key Points to Remember:
- HCF is the largest number that divides into both numbers - look for the intersection in Venn diagrams
- LCM is the smallest number that both original numbers divide into - use all factors from the Venn diagram
- Always express numbers as products of prime factors first
- Use index notation (powers) to write your final answers clearly
- In exams, show your working clearly using factor trees and Venn diagrams