Find the LCM and HCF of the following pair of integers and verify that LCM ` xx`HCF = Product of the two numbers
26 and 91

Answer

To find the LCM and HCF of the integers 26 and 91, we will follow these steps:

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

- **For 26**:
- 26 is divisible by 2 (the smallest prime number).
- \( 26 \div 2 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 26 is:
\[ 26 = 2^1 \times 13^1 \]

- **For 91**:
- 91 is not divisible by 2 (it is odd).
- The next prime number is 3 (91 is not divisible by 3).
- The next prime number is 5 (91 is not divisible by 5).
- The next prime number is 7.
- \( 91 \div 7 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 91 is:
\[ 91 = 7^1 \times 13^1 \]

### 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 between 26 and 91 is 13.
- Therefore, the HCF is:
\[ \text{HCF} = 13^1 = 13 \]

### 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 the prime factor 2: highest power is \( 2^1 \)
- For the prime factor 7: highest power is \( 7^1 \)
- For the prime factor 13: highest power is \( 13^1 \)

Thus, the LCM is:
\[ \text{LCM} = 2^1 \times 7^1 \times 13^1 = 2 \times 7 \times 13 \]
Calculating this gives:
\[ 2 \times 7 = 14 \]
\[ 14 \times 13 = 182 \]
So, the LCM is:
\[ \text{LCM} = 182 \]

### Step 4: Verification
Now, we verify that the product of LCM and HCF equals the product of the two numbers.

- Product of the two numbers:
\[ 26 \times 91 = 2366 \]

- Product of LCM and HCF:
\[ \text{LCM} \times \text{HCF} = 182 \times 13 \]
Calculating this gives:
\[ 182 \times 13 = 2366 \]

Since both products are equal:
\[ \text{LCM} \times \text{HCF} = 2366 = 26 \times 91 \]

### Final Answer
- HCF = 13
- LCM = 182