Find the L.C.M. and H.C.F. of the following pairs of integers and verify that `L.C.MxxH.C.F`=product of the two numbers 26 and 91

Answer

To find the L.C.M. (Lowest Common Multiple) and H.C.F. (Highest Common Factor) of the integers 26 and 91, we will follow these steps:

### Step 1: Find the Prime Factorization of Each Number
- **Prime Factorization of 26**:
- 26 can be divided by 2 (the smallest prime number).
- \( 26 \div 2 = 13 \)
- So, the prime factorization of 26 is \( 2 \times 13 \).

- **Prime Factorization of 91**:
- 91 can be divided by 7 (the smallest prime number that fits).
- \( 91 \div 7 = 13 \)
- So, the prime factorization of 91 is \( 7 \times 13 \).

### Step 2: Determine the H.C.F.
- The H.C.F. is found by taking the product of the lowest powers of common prime factors.
- The common prime factor between 26 and 91 is 13.
- Thus, the H.C.F. of 26 and 91 is \( 13 \).

### Step 3: Determine the L.C.M.
- The L.C.M. is found by taking the product of the highest powers of all prime factors present in both numbers.
- The prime factors are 2, 7, and 13.
- Therefore, the L.C.M. is calculated as follows:
- \( L.C.M. = 2^1 \times 7^1 \times 13^1 \)
- \( L.C.M. = 2 \times 7 \times 13 \)
- First, calculate \( 2 \times 7 = 14 \).
- Then, \( 14 \times 13 = 182 \).
- Thus, the L.C.M. of 26 and 91 is \( 182 \).

### Step 4: Verify the Relationship \( L.C.M. \times H.C.F. = \text{Product of the two numbers} \)
- Calculate the product of the two numbers:
- \( 26 \times 91 = 2366 \).
- Now calculate \( L.C.M. \times H.C.F. \):
- \( 182 \times 13 = 2366 \).
- Since both products are equal, we have verified that \( L.C.M. \times H.C.F. = 26 \times 91 \).

### Final Results:
- H.C.F. of 26 and 91 is \( 13 \).
- L.C.M. of 26 and 91 is \( 182 \).
- Verification: \( 182 \times 13 = 2366 \) which equals \( 26 \times 91 \).