Find the HCF and the LCM of 306 and 657, and verify that HCF × LCM = product of the two numbers.

HCF = 9, LCM = 22338, and 9 × 22338 = 201042 = 306 × 657.

Find the HCF and the LCM of $306$ and $657$ , and verify that HCF $\times$ LCM $=$ product of the two numbers.

medium 3 marks Class 10 · Mathematics · Real Numbers

Answer

HCF

= 9

, LCM

= 22338

, and

9 \times 22338 = 201042 = 306 \times 657

The correct answer is HCF $= 9$ , LCM $= 22338$ , and $9 \times 22338 = 201042 = 306 \times 657$ . — And the product of the two numbers is $306 \times 657 = 201042$ . The two agree, so the result is verified.

Step-by-step solution

Write each number as a product of primes. $306 = 2 \times 3^{2} \times 17$ and $657 = 3^{2} \times 73$ .
The HCF is the product of the common prime factors, each taken with the smallest power it appears with. Only $3^{2}$ is common, so HCF $= 3^{2} = 9$ .
The LCM is the product of all prime factors, each taken with the greatest power: LCM $= 2 \times 3^{2} \times 17 \times 73 = 22338$ .
Now check the property. HCF $\times$ LCM $= 9 \times 22338 = 201042$ .
And the product of the two numbers is $306 \times 657 = 201042$ . The two agree, so the result is verified.

Common mistake: Taking the highest power for the HCF and the lowest for the LCM. It is the other way round: the HCF takes the smallest power, and only of the primes common to both numbers.

Concept tested: Relationship between HCF and LCM

Published 19 August 2026 · How this solution was prepared and checked

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