MP Board · Class 10 · Mathematics · Real NumbersFind the HCF and LCM of 96 and 404 by the prime factorization method and verify that HCF × LCM = product of the two numbers.

To find the HCF and LCM of 96 and 404, we follow these steps using the Fundamental Theorem of Arithmetic:

Step 1: Prime Factorization of 96 96 = 2 × 2 × 2 × 2 × 2 × 3 96 = $2^5 \times 3^1$

Step 2: Prime Factorization of 404 404 = 2 × 2 × 101 404 = $2^2 \times 101^1$ (Note: 101 is a prime number)

Step 3: Finding the HCF (Highest Common Factor)\nThe HCF is the product of the smallest power of each common prime factor.\nCommon prime factor = 2\nSmallest power of 2 = $2^2$\nTherefore, HCF(96, 404) = $2^2 = 4$

Step 4: Finding the LCM (Least Common Multiple)\nThe LCM is the product of the greatest power of each prime factor involved in the numbers.\nLCM(96, 404) = $2^5 \times 3^1 \times 101^1$\nLCM(96, 404) = 32 × 3 × 101\nLCM(96, 404) = 96 × 101 = 9696

Step 5: Verification of the Relationship\nRelationship: HCF × LCM = Product of the two numbers\nLHS = HCF × LCM = 4 × 9696 = 38784\nRHS = Product of numbers = 96 × 404 = 38784\nSince LHS = RHS (38784 = 38784), the relationship is verified.