LCM anda HCF of two four digits number are 19261 and 103. Find the sum of these two numbers?

Options

A. 2884
B. 4296
C. 2464
D. 1996

Answer

To find the sum of the two four-digit numbers given their LCM and HCF, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- LCM = 19261
- HCF = 103

2. **Use the Relationship Between LCM, HCF, and the Two Numbers:**
The relationship is given by the formula:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Let the two numbers be \( x \) and \( y \). Therefore, we can write:
\[ x \times y = \text{LCM} \times \text{HCF} = 19261 \times 103 \]

3. **Calculate the Product of the Two Numbers:**
\[ x \times y = 19261 \times 103 = 1989993 \]

4. **Express the Numbers in Terms of HCF:**
Since the HCF is 103, we can express the two numbers as:
\[ x = 103a \quad \text{and} \quad y = 103b \]
where \( a \) and \( b \) are co-prime integers.

5. **Substitute into the Product Equation:**
Substitute \( x \) and \( y \) into the product equation:
\[ (103a) \times (103b) = 1989993 \]
Simplifying gives:
\[ 103^2 \times (a \times b) = 1989993 \]

6. **Calculate \( 103^2 \):**
\[ 103^2 = 10609 \]

7. **Find \( a \times b \):**
Now, substitute \( 10609 \) into the equation:
\[ 10609 \times (a \times b) = 1989993 \]
Dividing both sides by \( 10609 \):
\[ a \times b = \frac{1989993}{10609} = 187 \]

8. **Find the Factor Pairs of 187:**
The pairs of factors of 187 are:
- \( (1, 187) \)
- \( (11, 17) \)

9. **Check Which Pairs Yield Four-Digit Numbers:**
- For the pair \( (1, 187) \):
\[ x = 103 \times 1 = 103 \quad \text{(not a four-digit number)} \]
\[ y = 103 \times 187 = 19261 \quad \text{(not a four-digit number)} \]
- For the pair \( (11, 17) \):
\[ x = 103 \times 11 = 1133 \quad \text{(not a four-digit number)} \]
\[ y = 103 \times 17 = 1751 \quad \text{(not a four-digit number)} \]

10. **Calculate the Sum of the Two Numbers:**
Now, we can calculate the sum of the two four-digit numbers:
\[ \text{Sum} = 1133 + 1751 = 2884 \]

### Final Answer:
The sum of the two four-digit numbers is **2884**.