medium difficulty7 min

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.

In one line: For HCF, keep the common primes and trim the powers.

Practice This Concept

Understand it

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.

Example

36 = 2² × 3² and 48 = 2⁴ × 3, so the HCF is 2² × 3 = 12.

Formula and rule

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

See it worked out

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

Worked example

Find the HCF of 36 and 48.

1First write 36 = 2² × 3² and 48 = 2⁴ × 3.
2The common primes are 2 and 3.
3Use the smaller powers: 2² and 3¹.
4So HCF = 2² × 3 = 12.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In 54 = 2 × 3³ and 72 = 2³ × 3², what should be taken for the HCF?

Think first, then show answer

Take the common primes 2 and 3 with the smaller powers, so the HCF is 2 × 3² = 18.

Common wrong answer

Using 2⁴ × 3² for 36 and 48.

Why it loses marks: Those are the highest powers, not the common minimum powers.

Don't mix this up

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

Exam tip

For HCF, remember the phrase: common primes with minimum 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 HCF from prime factorisation, or identify which powers should be chosen.

1
Enough for 1 mark
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
Add for 2 marks
Example: 36 = 2² × 3² and 48 = 2⁴ × 3, so the HCF is 2² × 3 = 12.
3
Add for 3 marks
The HCF must divide both numbers fully, so it can only use the prime factors that are common to both numbers.

MCQ Quiz

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