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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Using product minus HCF is wrong because this relation works by multiplication and division, not subtraction.

Definition

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

Example

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

Rule to remember

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

Memory hook

Product splits into HCF and LCM by division.

Examples and method

Worked example

For 24 and 36, HCF = 12. Their product is 864. So LCM = 864 / 12 = 72.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Using product minus HCF is wrong because this relation works by multiplication and division, not subtraction.

Exam tip

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

Quick check

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

Their LCM is 960 divided by 8, which is 120.

Answer writing and exam use

1-mark answer

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

2-mark answer

For two positive integers a and b, the product of the numbers equals the product of their HCF and LCM. a x b = HCF(a,b) x LCM(a,b). If the numbers are 18 and 30, then 18 x 30 = 6 x 90.

3-mark answer

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. a x b = HCF(a,b) x LCM(a,b). For 24 and 36, HCF = 12. Their product is 864. So LCM = 864 / 12 = 72. Find the LCM when the HCF and product are given, or verify a pair after factorisation. Using product minus HCF is wrong because this relation works by multiplication and division, not subtraction.
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