HCF and LCM of two 3 - digit numbers are 23 and 1771 respectively. Find the sum of their numbers:

Options

A. 414
B. 391
C. 460
D. 322

Answer

To find the sum of two 3-digit numbers given their HCF and LCM, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- HCF (Highest Common Factor) = 23
- LCM (Lowest Common Multiple) = 1771

2. **Use the relationship between HCF, LCM, and the product of the two numbers**:
The relationship is given by the formula:
\[ \text{HCF} \times \text{LCM} = a \times b \]
where \(a\) and \(b\) are the two numbers we need to find.

3. **Substitute the known values into the formula**:
\[ 23 \times 1771 = a \times b \]

4. **Calculate the product**:
\[ 23 \times 1771 = 40733 \]
So, we have:
\[ a \times b = 40733 \]

5. **Express the numbers in terms of HCF**:
Since the HCF is 23, we can express the two numbers as:
\[ a = 23m \quad \text{and} \quad b = 23n \]
where \(m\) and \(n\) are coprime integers.

6. **Substitute into the product equation**:
\[ (23m) \times (23n) = 40733 \]
Simplifying gives:
\[ 529mn = 40733 \]

7. **Solve for \(mn\)**:
\[ mn = \frac{40733}{529} = 77 \]

8. **Find pairs of \(m\) and \(n\)**:
The pairs of integers \(m\) and \(n\) that multiply to 77 are:
- (1, 77)
- (7, 11)

9. **Calculate the corresponding numbers**:
- For \(m = 1\) and \(n = 77\):
\[ a = 23 \times 1 = 23 \quad \text{and} \quad b = 23 \times 77 = 1771 \quad \text{(not a 3-digit number)} \]
- For \(m = 7\) and \(n = 11\):
\[ a = 23 \times 7 = 161 \quad \text{and} \quad b = 23 \times 11 = 253 \]

10. **Check if both numbers are 3-digit numbers**:
Both 161 and 253 are 3-digit numbers.

11. **Calculate the sum of the two numbers**:
\[ 161 + 253 = 414 \]

### Final Answer:
The sum of the two numbers is **414**.

---