Find the HCF and LCM of $84$ and $150$ using prime factorisation, and verify that HCF $\times$ LCM equals the product $84 \times 150$ .

medium 2 marks Class 10 · Mathematics · Real Numbers

Answer

HCF = 6, LCM = 2100, and

6 \times 2100 = 84 \times 150 = 12600

The correct answer is HCF = 6, LCM = 2100, and $6 \times 2100 = 84 \times 150 = 12600$ . — Both are equal, confirming the property.

Step-by-step solution

Prime factorise each number:
$84 = 2^{2} \times 3 \times 7$ ;
$150 = 2 \times 3 \times 5^{2}$ .
For the HCF, take the lowest power of each common prime:
Common primes are $2$ and $3$ ;
Lowest powers: $2^{1}$ and $3^{1}$ ;
Thus HCF $= 2^{1} \times 3^{1} = 6$ .
For the LCM, take the highest power of every prime appearing:
$2^{2}$ , $3^{1}$ , $5^{2}$ , $7^{1}$ ;
Thus LCM $= 2^{2} \times 3^{1} \times 5^{2} \times 7^{1} = 4 \times 3 \times 25 \times 7 = 2100$ .
Now verify the product property:
HCF $\times$ LCM $= 6 \times 2100 = 12600$ ;
Product of the numbers $= 84 \times 150 = 12600$ ;
Both are equal, confirming the property.

Common mistake: Using the lowest power of a prime for LCM or the highest power for HCF, which reverses the required rule.

Concept tested: Prime factorisation for HCF and LCM