CBSE · Class 10 · Mathematics · Real NumbersFind the HCF and LCM of 336 and 54 by the prime factorization method, and verify that $\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$.

To find the HCF and LCM of 336 and 54 using the prime factorization method, we first find the prime factors of each number. \nStep 1: Find the prime factorization of 336. $$336 = 2 \times 168 = 2 \times 2 \times 84 = 2 \times 2 \times 2 \times 42 = 2 \times 2 \times 2 \times 2 \times 21 = 2^4 \times 3^1 \times 7^1$$ \nStep 2: Find the prime factorization of 54. $$54 = 2 \times 27 = 2 \times 3 \times 3 \times 3 = 2^1 \times 3^3$$ \nStep 3: Calculate the HCF.\nThe HCF is the product of the smallest power of each common prime factor in the numbers. The common prime factors are 2 and 3. $$\text{HCF}(336, 54) = 2^1 \times 3^1 = 2 \times 3 = 6$$ \nStep 4: Calculate the LCM.\nThe LCM is the product of the greatest power of each prime factor involved in the numbers. $$\text{LCM}(336, 54) = 2^4 \times 3^3 \times 7^1 = 16 \times 27 \times 7 = 432 \times 7 = 3024$$ \nStep 5: Verify the relation $\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$. $$\text{HCF} \times \text{LCM} = 6 \times 3024 = 18144$$ $$\text{Product of the two numbers} = 336 \times 54 = 18144$$ \nSince $\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$ ($18144 = 18144$), the given relation is verified.