Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the twogiven numbers

1. Break Down the Problem

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 404 and 96, we will follow these steps:

• Step 1: Find the prime factorization of both numbers.
• Step 2: Calculate the HCF using the prime factorizations.
• Step 3: Calculate the LCM using the prime factorizations.
• Step 4: Verify that the product of HCF and LCM equals the product of the two given numbers.

2. Relevant Concepts

• HCF: The highest number that divides both numbers without leaving a remainder.
• LCM: The smallest number that is a multiple of both numbers.
• Prime Factorization: Expressing numbers as the product of prime numbers.

3. Analysis and Detail

Step 1: Prime Factorization

• 404: $404 = 2^2 \times 101^1$
• 96: $96 = 2^5 \times 3^1$

Step 2: Calculate HCF

HCF is found by taking the lowest power of common prime factors:

• Common prime factor: $2$
• HCF: $2^{\min(2, 5)} = 2^2 = 4$

Step 3: Calculate LCM

LCM is found by taking the highest power of all prime factors:

• LCM: $2^{\max(2, 5)} \times 3^{\max(0, 1)} \times 101^{\max(1, 0)}$
• LCM: $2^5 \times 3^1 \times 101^1$

Calculating:

• $2^5 = 32$
• $3^1 = 3$
• $101^1 = 101$

Thus, $\text{LCM} = 32 \times 3 \times 101 = 32 \times 303 = 9696$

Step 4: Verify $\text{HCF} \times \text{LCM} = \text{Product of the numbers}$

Calculating the product:

• Product of the numbers: $404 \times 96 = 38784$
• HCF × LCM: $4 \times 9696 = 38784$

4. Verify and Summarize

Both calculations confirm that:

• HCF = 4
• LCM = 9696
• HCF × LCM = 38784, which equals the product of the numbers (404 × 96).

Final Answer

• HCF: 4
• LCM: 9696
• Verification: $\text{HCF} \times \text{LCM} = 38784$ (which is $404 \times 96$) shows that the calculation is correct.