Find the LCM and HCF of the following pair of integers and verify that LCM ` xx`HCF = Product of the two numbers
510 and 92

Answer

To find the LCM and HCF of the integers 510 and 92, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of both numbers.

**For 92:**
- Divide by 2: \( 92 \div 2 = 46 \)
- Divide by 2 again: \( 46 \div 2 = 23 \)
- 23 is a prime number.

So, the prime factorization of 92 is:
\[ 92 = 2^2 \times 23^1 \]

**For 510:**
- Divide by 2: \( 510 \div 2 = 255 \)
- Divide by 3: \( 255 \div 3 = 85 \)
- Divide by 5: \( 85 \div 5 = 17 \)
- 17 is a prime number.

So, the prime factorization of 510 is:
\[ 510 = 2^1 \times 3^1 \times 5^1 \times 17^1 \]

### Step 2: Finding HCF
The HCF (Highest Common Factor) is found by taking the lowest power of all common prime factors.

- The common prime factor is 2.
- The lowest power of 2 is \( 2^1 \).

Thus, the HCF is:
\[ \text{HCF} = 2^1 = 2 \]

### Step 3: Finding LCM
The LCM (Lowest Common Multiple) is found by taking the highest power of all prime factors present in either number.

- For 2: \( 2^2 \) (from 92)
- For 3: \( 3^1 \) (from 510)
- For 5: \( 5^1 \) (from 510)
- For 17: \( 17^1 \) (from 510)
- For 23: \( 23^1 \) (from 92)

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 17^1 \times 23^1 \]

Calculating this:
\[ \text{LCM} = 4 \times 3 \times 5 \times 17 \times 23 \]

Calculating step-by-step:
1. \( 4 \times 3 = 12 \)
2. \( 12 \times 5 = 60 \)
3. \( 60 \times 17 = 1020 \)
4. \( 1020 \times 23 = 23460 \)

So, the LCM is:
\[ \text{LCM} = 23460 \]

### Step 4: Verification
Now, we verify that \( \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \).

Calculating the product of the two numbers:
\[ 510 \times 92 = 46920 \]

Now, calculate \( \text{LCM} \times \text{HCF} \):
\[ 23460 \times 2 = 46920 \]

Since both products are equal, we have verified the relationship.

### Final Results
- HCF = 2
- LCM = 23460