Use prime factors to find GCF and LCM
Write a prime factorisation · Choose shared factors for the greatest common factor · Choose all required factors for the least common multiple
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Factors, multiples and prime numbers
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Write a prime factorisation
- Choose shared factors for the greatest common factor
- Choose all required factors for the least common multiple
Factor and multiple
A factor divides a whole number exactly; a multiple is an integer times that number. Break a positive integer greater than 1 into prime factors, retaining repeated factors: 36 = 2² × 3².
Two different selections
The GCF uses the smaller exponent of each prime shared by all numbers. The LCM uses the larger exponent of every prime needed by any number. GCF is useful for equal groups or simplifying fractions; LCM is useful for repeating schedules or common denominators.
For 24 = 2³ × 3 and 36 = 2² × 3², GCF = 2² × 3 = 12 and LCM = 2³ × 3² = 72. Check that the GCF divides both and that the LCM is a multiple of both.
Sources for this lesson
Pause and explain
Which prime powers form the LCM?
Worked example
Find GCF and LCM of 18 and 30.
Show the worked solution
- 18 = 2 × 3²; 30 = 2 × 3 × 5.
- Shared minimum powers give 2 × 3 = 6.
- Maximum powers give 2 × 3² × 5 = 90.
Answer GCF 6; LCM 90.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Factorise 48 into primes.
Give me a hint
Keep dividing by a prime.
Compare my reasoning
- 48 = 2 × 24 = 2² × 12.
- 12 = 2² × 3, so 48 = 2⁴ × 3.
2⁴ × 3.
Look for these in your work
- End with prime factors
- Multiply back to check
Find GCF and LCM of 40 and 60.
Give me a hint
Use 40 = 2³ × 5 and 60 = 2² × 3 × 5.
Compare my reasoning
- GCF = 2² × 5 = 20.
- LCM = 2³ × 3 × 5 = 120.
GCF 20; LCM 120.
Look for these in your work
- Use minimum shared powers
- Use maximum required powers
Two alarms ring every 12 and 18 minutes, together at 09:00. When next together?
Give me a hint
Find the least positive common multiple.
Compare my reasoning
- 12 = 2² × 3 and 18 = 2 × 3².
- LCM = 36 minutes, added to 09:00.
09:36.
Look for these in your work
- Choose LCM for a repeated schedule
- Require the next positive interval
Common mistakes
- Do not confuse greatest common factor with greatest number on a factor tree.
- Keep repeated prime factors.
What can you explain now?
Which prime powers form the LCM?
Compare with the explanation
The greatest required powers of every prime present.
The multiple must contain enough of each prime to be divisible by all numbers.
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Make it stick
Which prime powers form the LCM?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.