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Prime factorisation for HCF

To find the HCF by prime factorisation, write both numbers as products of primes and take only the common prime factors with the smallest powers.

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Student-friendly explanation

The HCF must divide both numbers fully, so it can only use the prime factors that are common to both numbers. For each common prime, we choose the smaller power, because the smaller count is the limit shared by both numbers.

How to write this in exams

  1. 1

    Start with the exact idea

    To find the HCF by prime factorisation, write both numbers as products of primes and take only the common prime factors with the smallest powers.

  2. 2

    Then show how to use it

    1. Prime-factorise both numbers. 2. Mark the prime factors common to both. 3. Choose the smaller power of each common prime. 4. Multiply those selected factors.

  3. 3

    Add one concrete example

    36 = 2^2 x 3^2 and 48 = 2^4 x 3, so the HCF is 2^2 x 3 = 12.

  4. 4

    Avoid this incomplete answer

    Using 2^4 x 3^2 for 36 and 48 is wrong because those are the highest powers, not the common minimum powers.

Definition

To find the HCF by prime factorisation, write both numbers as products of primes and take only the common prime factors with the smallest powers.

Example

36 = 2^2 x 3^2 and 48 = 2^4 x 3, so the HCF is 2^2 x 3 = 12.

Rule to remember

HCF = product of common prime factors with the least powers.

Memory hook

For HCF, keep the common primes and trim the powers.

Examples and method

Worked example

Find the HCF of 36 and 48. First write 36 = 2^2 x 3^2 and 48 = 2^4 x 3. The common primes are 2 and 3. Use the smaller powers: 2^2 and 3^1. So HCF = 2^2 x 3 = 12.

Method to apply

1. Prime-factorise both numbers. 2. Mark the prime factors common to both. 3. Choose the smaller power of each common prime. 4. Multiply those selected factors.

Diagram support

A two-column factor list with the common primes highlighted makes the method easy to follow.

How CBSE asks it

Find the HCF from prime factorisation, or identify which powers should be chosen.

Avoid common mistakes

Common confusion

Students often take the largest powers of common primes, which gives the LCM idea instead of the HCF.

Common wrong answer

Using 2^4 x 3^2 for 36 and 48 is wrong because those are the highest powers, not the common minimum powers.

Exam tip

For HCF, remember the phrase: common primes with minimum powers.

Quick check

In 54 = 2 x 3^3 and 72 = 2^3 x 3^2, what should be taken for the HCF?

Take the common primes 2 and 3 with the smaller powers, so the HCF is 2 x 3^2 = 18.

Answer writing and exam use

1-mark answer

To find the HCF by prime factorisation, write both numbers as products of primes and take only the common prime factors with the smallest powers.

2-mark answer

To find the HCF by prime factorisation, write both numbers as products of primes and take only the common prime factors with the smallest powers. HCF = product of common prime factors with the least powers. 36 = 2^2 x 3^2 and 48 = 2^4 x 3, so the HCF is 2^2 x 3 = 12.

3-mark answer

The HCF must divide both numbers fully, so it can only use the prime factors that are common to both numbers. For each common prime, we choose the smaller power, because the smaller count is the limit shared by both numbers. HCF = product of common prime factors with the least powers. Find the HCF of 36 and 48. First write 36 = 2^2 x 3^2 and 48 = 2^4 x 3. The common primes are 2 and 3. Use the smaller powers: 2^2 and 3^1. So HCF = 2^2 x 3 = 12. Find the HCF from prime factorisation, or identify which powers should be chosen. Using 2^4 x 3^2 for 36 and 48 is wrong because those are the highest powers, not the common minimum powers.
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