medium difficulty7 min

Prime factorisation for LCM

To find the LCM by prime factorisation, write both numbers as primes and take every prime that appears in either number, using the highest power needed.

In one line: For LCM, build upward with the biggest prime powers.

Practice This Concept

Understand it

The LCM is the smallest common multiple, so it must contain enough prime factors to cover both numbers completely. That is why we choose the greatest power of each prime appearing in either number, not the smallest one.

Example

36 = 2² × 3² and 48 = 2⁴ × 3, so the LCM is 2⁴ × 3² = 144.

Formula and rule

LCM = product of all prime factors with the greatest powers.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the LCM of 36 and 48.

1Write 36 = 2² × 3² and 48 = 2⁴ × 3.
2Take the larger powers: 2⁴ and 3².
3So LCM = 2⁴ × 3² = 144.

Method you can reuse

1Prime-factorise both numbers.
2List all primes that appear.
3Choose the greater power for each prime.
4Multiply the selected factors.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In 45 = 3² × 5 and 60 = 2² × 3 × 5, what powers should be taken for the LCM?

Think first, then show answer

Take 2², 3², and 5¹, because the LCM needs the highest power of each prime appearing in either number.

Common wrong answer

Using 2² × 3 = 12 for 36 and 48.

Why it loses marks: It uses minimum powers and gives an HCF-like result.

Don't mix this up

Students take the smallest powers and end up finding the HCF instead of the LCM.

Exam tip

For LCM, remember the phrase: all primes with maximum powers.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find the least common multiple using prime factors, or compare HCF and LCM steps.

1
Enough for 1 mark
To find the LCM by prime factorisation, write both numbers as primes and take every prime that appears in either number, using the highest power needed.
2
Add for 2 marks
Example: 36 = 2² × 3² and 48 = 2⁴ × 3, so the LCM is 2⁴ × 3² = 144.
3
Add for 3 marks
The LCM is the smallest common multiple, so it must contain enough prime factors to cover both numbers completely.

MCQ Quiz

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