Find the HCF and LCM of $84$ and $126$ using prime factorisation, and verify that $\text{HCF}\times \text{LCM}$ equals the product of the two numbers.

Step-by-step solution

1. Write the prime factorisation of each number: $84=2^2\times 3\times 7$, $126=2\times 3^2\times 7$.
2. Identify the common prime factors with the smallest exponents: $2^{\min (2,1)}=2$, $3^{\min (1,2)}=3$, $7^{\min (1,1)}=7$.
3. Multiply these common factors to obtain the HCF: $2\times 3\times 7=42$.
4. For the LCM, take each prime factor with the largest exponent: $2^{\max (2,1)}=2^2$, $3^{\max (1,2)}=3^2$, $7^{\max (1,1)}=7$.
5. Multiply them: $2^2\times 3^2\times 7=4\times 9\times 7=252$.
6. Check the product rule: $\text{HCF}\times \text{LCM}=42\times 252=10584$ and $84\times 126=10584$, confirming the rule.

Concept tested: HCF and LCM via prime factorisation

Published 22 August 2026 · How this solution was prepared and checked

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