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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    36 = 2^2 x 3^2 and 48 = 2^4 x 3, so the LCM is 2^4 x 3^2 = 144.

  4. 4

    Avoid this incomplete answer

    Using 2^2 x 3 = 12 for 36 and 48 is wrong because it uses minimum powers and gives an HCF-like result.

Definition

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.

Example

36 = 2^2 x 3^2 and 48 = 2^4 x 3, so the LCM is 2^4 x 3^2 = 144.

Rule to remember

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

Memory hook

For LCM, build upward with the biggest prime powers.

Examples and method

Worked example

Find the LCM of 36 and 48. Write 36 = 2^2 x 3^2 and 48 = 2^4 x 3. Take the larger powers: 2^4 and 3^2. So LCM = 2^4 x 3^2 = 144.

Method to apply

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

Diagram support

A prime-power comparison table helps students choose the largest exponent for each prime.

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Using 2^2 x 3 = 12 for 36 and 48 is wrong because it uses minimum powers and gives an HCF-like result.

Exam tip

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

Quick check

In 45 = 3^2 x 5 and 60 = 2^2 x 3 x 5, what powers should be taken for the LCM?

Take 2^2, 3^2, and 5^1, because the LCM needs the highest power of each prime appearing in either number.

Answer writing and exam use

1-mark answer

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-mark answer

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. LCM = product of all prime factors with the greatest powers. 36 = 2^2 x 3^2 and 48 = 2^4 x 3, so the LCM is 2^4 x 3^2 = 144.

3-mark answer

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. LCM = product of all prime factors with the greatest powers. Find the LCM of 36 and 48. Write 36 = 2^2 x 3^2 and 48 = 2^4 x 3. Take the larger powers: 2^4 and 3^2. So LCM = 2^4 x 3^2 = 144. Find the least common multiple using prime factors, or compare HCF and LCM steps. Using 2^2 x 3 = 12 for 36 and 48 is wrong because it uses minimum powers and gives an HCF-like result.
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