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CraftExam
medium difficulty6 min

Relation between HCF and LCM

For two positive integers a and b, the product of the numbers equals the product of their HCF and LCM.

In one line: Product splits into HCF and LCM by division.

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Understand it

This relation is a quick check in exam questions. Once the HCF is known, the LCM can be found quickly, and the relation also helps verify whether a final answer is reasonable. It works for two positive integers.

Example

If the numbers are 18 and 30, then 18 × 30 = 6 × 90.

Formula and rule

a x b = HCF(a,b) x LCM(a,b).

See it worked out

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

Worked example

For 24 and 36, HCF = 12.

  1. 1Their product is 864.
  2. 2So LCM = 864 / 12 = 72.

Method you can reuse

  1. 1Write the product of the two numbers.
  2. 2Divide by the HCF.
  3. 3The result is the LCM.
  4. 4Check that HCF x LCM matches the product.

Visual tip

A simple balance-style box showing product on one side and HCF and LCM on the other helps memory.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If the HCF of two numbers is 8 and their product is 960, what is their LCM?

Think first, then show answer
Their LCM is 960 divided by 8, which is 120.

Common wrong answer

Using product minus HCF.

Why it loses marks: This relation works by multiplication and division, not subtraction.

Don't mix this up

Students divide by the wrong value or try to apply the relation to more than two numbers at once.

Exam tip

Use this relation as a shortcut after finding either the HCF or the LCM by factorisation.

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 LCM when the HCF and product are given, or verify a pair after factorisation.
  1. 1

    Enough for 1 mark

    For two positive integers a and b, the product of the numbers equals the product of their HCF and LCM.

  2. 2

    Add for 2 marks

    Example: If the numbers are 18 and 30, then 18 × 30 = 6 × 90.

  3. 3

    Add for 3 marks

    Once the HCF is known, the LCM can be found quickly, and the relation also helps verify whether a final answer is reasonable.

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