Find HCF and LCM of 404 and 96 and verify that `HCF times LCM `=Product of the two given numbers

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 404 and 96, and to verify that HCF × LCM = Product of the two numbers, we can follow these steps:

### Step 1: Prime Factorization of 404
To find the prime factorization of 404:
- 404 is even, so we divide by 2:
\[ 404 \div 2 = 202 \]
- 202 is also even, so we divide by 2 again:
\[ 202 \div 2 = 101 \]
- 101 is a prime number.

Thus, the prime factorization of 404 is:
\[ 404 = 2^2 \times 101^1 \]

### Step 2: Prime Factorization of 96
Now, we find the prime factorization of 96:
- 96 is even, so we divide by 2:
\[ 96 \div 2 = 48 \]
- 48 is even, so we divide by 2 again:
\[ 48 \div 2 = 24 \]
- 24 is even, so we divide by 2 again:
\[ 24 \div 2 = 12 \]
- 12 is even, so we divide by 2 again:
\[ 12 \div 2 = 6 \]
- 6 is even, so we divide by 2 again:
\[ 6 \div 2 = 3 \]
- 3 is a prime number.

Thus, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

### Step 3: Finding HCF
To find the HCF, we take the lowest power of each common prime factor:
- For the prime factor 2: The minimum power is \(2^2\).
- For the prime factor 101: It is not common in 96.
- For the prime factor 3: It is not common in 404.

Thus, the HCF is:
\[ HCF = 2^2 = 4 \]

### Step 4: Finding LCM
To find the LCM, we take the highest power of each prime factor:
- For the prime factor 2: The maximum power is \(2^5\).
- For the prime factor 101: The maximum power is \(101^1\).
- For the prime factor 3: The maximum power is \(3^1\).

Thus, the LCM is:
\[ LCM = 2^5 \times 3^1 \times 101^1 \]
Calculating this:
\[ = 32 \times 3 \times 101 \]
\[ = 96 \times 101 = 9696 \]

### Step 5: Verification
Now we verify that HCF × LCM = Product of the two numbers:
- HCF = 4
- LCM = 9696
- Product of the two numbers = 404 × 96

Calculating the product:
\[ 404 \times 96 = 38784 \]

Now, calculating HCF × LCM:
\[ 4 \times 9696 = 38784 \]

Since both sides are equal:
\[ HCF \times LCM = Product \text{ of the two numbers} \]
Thus, the verification is complete.

### Final Results
- HCF = 4
- LCM = 9696