Highest common factor (HCF) and lowest common multiple (LCM)

What you'll learn

This revision guide covers finding the highest common factor (HCF) and lowest common multiple (LCM) of numbers, essential skills tested across Foundation and Higher tier AQA GCSE Mathematics papers. You'll master multiple methods including listing, prime factorisation, and Venn diagrams, and apply these techniques to solve real-world problems involving time, measurement, and patterns.

Key terms and definitions

Factor — a whole number that divides exactly into another number with no remainder.

Multiple — the result of multiplying a number by any positive integer.

Highest Common Factor (HCF) — the largest number that divides exactly into two or more numbers.

Lowest Common Multiple (LCM) — the smallest number that is a multiple of two or more numbers.

Prime number — a number greater than 1 that has exactly two factors: 1 and itself.

Prime factorisation — expressing a number as a product of its prime factors.

Product of prime factors — writing a number in the form 2^a × 3^b × 5^c × ... where the bases are prime numbers.

Common factor — a number that is a factor of two or more numbers.

Core concepts

Understanding factors and multiples

Factors divide into a number exactly, while multiples are created by multiplication.

For example, consider the number 12:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Multiples of 12: 12, 24, 36, 48, 60, ...

Key points:

• Every number has a finite number of factors
• Every number has an infinite number of multiples
• 1 is a factor of every number
• Every number is a factor of itself

When working with two or more numbers, you identify common factors (factors shared by all numbers) and common multiples (multiples shared by all numbers).

Finding the HCF using different methods

Method 1: Listing factors

This method works well for smaller numbers.

Steps:

1. List all factors of each number
2. Identify factors that appear in all lists
3. Select the largest common factor

Example: Find the HCF of 24 and 36

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors: 1, 2, 3, 4, 6, 12
• HCF = 12

Method 2: Prime factorisation

This is the most efficient method for larger numbers and is explicitly required in AQA specifications.

Steps:

1. Write each number as a product of prime factors
2. Identify prime factors common to all numbers
3. Multiply the common prime factors, using the lowest power of each

Example: Find the HCF of 72 and 90

72 = 2³ × 3² 90 = 2 × 3² × 5

Common prime factors: 2¹ and 3²

HCF = 2 × 3² = 2 × 9 = 18

Method 3: Venn diagrams

Venn diagrams provide a visual method particularly useful for solving HCF and LCM simultaneously.

Steps:

1. Write each number as a product of prime factors
2. Draw overlapping circles (one per number)
3. Place common prime factors in the overlap
4. Place remaining factors in the non-overlapping regions
5. HCF = product of numbers in the overlap

This method is covered in more detail in the worked examples section.

Finding the LCM using different methods

Method 1: Listing multiples

Suitable for smaller numbers only.

Steps:

1. List multiples of each number
2. Identify the smallest multiple that appears in all lists

Example: Find the LCM of 6 and 8

• Multiples of 6: 6, 12, 18, 24, 30, 36, 48, ...
• Multiples of 8: 8, 16, 24, 32, 40, 48, ...
• LCM = 24

Method 2: Prime factorisation

The most reliable method for all numbers.

Steps:

1. Write each number as a product of prime factors
2. For each prime factor that appears, take the highest power
3. Multiply these together

Example: Find the LCM of 72 and 90

72 = 2³ × 3² 90 = 2 × 3² × 5

Highest powers: 2³, 3², 5¹

LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360

Method 3: Venn diagrams

Using the same Venn diagram constructed for HCF:

1. Multiply all numbers in the diagram (both overlapping and non-overlapping regions)
2. This product equals the LCM

Prime factorisation techniques

To write a number as a product of prime factors, use a factor tree or division ladder.

Factor tree method:

1. Split the number into any pair of factors
2. Continue splitting until all branches end in prime numbers
3. Write the answer as a product of the prime factors

Example: 60

        60
       /  \
      6    10
     / \   / \
    2   3 2   5

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Division ladder method:

1. Divide by the smallest prime factor (usually 2)
2. Continue dividing by prime numbers until you reach 1
3. The prime factors are the divisors used

Example: 60

2 | 60
2 | 30
3 | 15
5 | 5
  | 1

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Working with three or more numbers

The same principles apply when finding HCF and LCM of three or more numbers.

For HCF with prime factorisation:

• Identify prime factors common to ALL numbers
• Use the lowest power of each common prime

For LCM with prime factorisation:

• List all prime factors that appear in ANY number
• Use the highest power of each prime

Example: Find HCF and LCM of 12, 18, and 30

12 = 2² × 3 18 = 2 × 3² 30 = 2 × 3 × 5

HCF: Common to all are 2¹ and 3¹ HCF = 2 × 3 = 6

LCM: Highest powers are 2², 3², 5¹ LCM = 4 × 9 × 5 = 180

Application problems

AQA exam questions often present HCF and LCM in context. Recognise the key scenarios:

HCF problems:

• Dividing or cutting things into equal-sized groups
• Finding the largest size of square tiles that fit exactly
• Sharing or distributing items equally
• Keywords: "largest," "maximum," "divide exactly"

LCM problems:

• Events happening repeatedly that coincide
• Pattern repetition
• Scheduling problems
• Keywords: "next time," "together again," "repeat," "coincide"

Example contexts:

• Bells ringing at different intervals
• Buses departing at different frequencies
• Cutting ribbon/wood into equal lengths with no waste
• Arranging items in rows with equal numbers

Worked examples

Example 1: Using Venn diagrams (Higher tier)

Question: Use a Venn diagram to find the HCF and LCM of 84 and 126. (3 marks)

Solution:

Step 1: Write as products of prime factors

• 84 = 2² × 3 × 7
• 126 = 2 × 3² × 7

Step 2: Draw Venn diagram with circles for 84 and 126

Place common factors in the overlap:

• Both have at least 2¹, 3¹, and 7¹
• Overlap contains: 2, 3, 7

Place remaining factors outside overlap:

• 84's circle also has: 2 (since 84 has 2² but only one 2 is in overlap)
• 126's circle also has: 3 (since 126 has 3² but only one 3 is in overlap)

Step 3: Calculate HCF and LCM

• HCF = product of overlap = 2 × 3 × 7 = 42 ✓ (1 mark)
• LCM = product of all values = 2 × 3 × 7 × 2 × 3 = 252 ✓ (2 marks)

Mark scheme: 1 mark for correct prime factorisation or Venn diagram; 1 mark for HCF = 42; 1 mark for LCM = 252

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Example 2: Real-world application (Foundation/Higher)

Question: Two bells ring at different intervals. Bell A rings every 45 seconds. Bell B rings every 60 seconds. Both bells ring together at 9:00 am. At what time will they next ring together? (3 marks)

Solution:

This is an LCM problem (events repeating and coinciding).

Step 1: Find LCM of 45 and 60 using prime factorisation

• 45 = 3² × 5
• 60 = 2² × 3 × 5

Step 2: Take highest powers of all primes

• LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 ✓ (1 mark)

Step 3: Interpret the answer

• The bells ring together every 180 seconds
• 180 seconds = 3 minutes ✓ (1 mark)
• Next time = 9:00 am + 3 minutes = 9:03 am ✓ (1 mark)

Mark scheme: 1 mark for LCM = 180; 1 mark for converting to minutes; 1 mark for correct time

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Example 3: Problem involving HCF (Foundation/Higher)

Question: A carpenter has two planks of wood. One is 96 cm long and the other is 144 cm long. He wants to cut both planks into pieces of equal length with no wood left over. What is the greatest possible length of each piece? (2 marks)

Solution:

This is an HCF problem (largest size that divides exactly).

Step 1: Find HCF of 96 and 144

• 96 = 2⁵ × 3
• 144 = 2⁴ × 3² ✓ (1 mark)

Step 2: Identify common factors with lowest powers

• Common: 2⁴ and 3¹
• HCF = 2⁴ × 3 = 16 × 3 = 48 cm ✓ (1 mark)

Answer: The greatest possible length is 48 cm.

Mark scheme: 1 mark for correct method (prime factorisation or listing factors); 1 mark for HCF = 48 cm

Common mistakes and how to avoid them

• Confusing HCF with LCM: Remember HCF is the largest factor (smaller number), while LCM is the smallest multiple (larger number). Check your answer makes sense—HCF ≤ both original numbers; LCM ≥ both original numbers.

• Using highest powers for HCF: When using prime factorisation for HCF, use the lowest power of common primes, not the highest. For LCM, use highest powers. A useful check: multiply HCF × LCM of two numbers = product of the two numbers.

• Missing prime factors: When constructing factor trees or division ladders, ensure you continue until all factors are prime. Remember 1 is not a prime number and should not appear in your product of primes.

• Not writing in index form: Always express repeated prime factors using indices (e.g., 2³ not 2 × 2 × 2) for clarity and to avoid errors, especially in longer calculations.

• Misidentifying problem type: Read questions carefully. "Largest," "maximum size," and "divide exactly" usually indicate HCF. "Next time together," "repeat," and "coincide" typically indicate LCM.

• Forgetting to convert units: In real-world problems, ensure all measurements use the same units before finding HCF or LCM. Convert your final answer to appropriate units (e.g., seconds to minutes) with clear working.

Exam technique for HCF and LCM questions

• Show full working for prime factorisation: Even if you can factorise mentally, draw a factor tree or division ladder. Method marks are available even if your final answer is incorrect, and this prevents arithmetic errors.

• Use index notation throughout: Questions explicitly asking for "product of prime factors" require index form (e.g., 2² × 3³). This is worth specific marks on the mark scheme—writing 2 × 2 × 3 × 3 × 3 loses marks.

• Identify question type early: Spend 5 seconds determining whether the question needs HCF or LCM before calculating. Underline key words like "largest," "maximum," "next time," or "together again" to avoid selecting the wrong value.

• Check answers are reasonable: After calculation, verify that HCF divides into both numbers exactly and LCM is divisible by both numbers. This catches most calculation errors and takes only seconds.

Quick revision summary

HCF is the largest number dividing exactly into all given numbers; LCM is the smallest number that all given numbers divide into. Use listing methods for small numbers, but prime factorisation is most reliable for larger numbers and exam questions. For HCF, multiply common prime factors using lowest powers; for LCM, multiply all prime factors using highest powers. Venn diagrams allow both to be found simultaneously. Recognise HCF in division/cutting problems and LCM in repeating/coinciding scenarios. Always show full working and use index notation.