The HCF of two numbers is 98 and their LCM is 2352. The sum of the numbers may be ?

Options

A. 1372
B. 2450
C. 1426
D. 1484

Answer

To solve the problem, we need to find two numbers whose Highest Common Factor (HCF) is 98 and their Least Common Multiple (LCM) is 2352. We will also find the sum of these two numbers.

### Step-by-Step Solution:

1. **Use the relationship between HCF, LCM, and the product of the numbers:**
The relationship states that:
\[ \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \]
Given that HCF = 98 and LCM = 2352, we can calculate the product of the two numbers.
\[ \text{Product} = 98 \times 2352 \]

2. **Calculate the product:**
\[ 98 \times 2352 = 230496 \]

3. **Factor the product to find possible pairs of numbers:**
We need to find pairs of factors of 230496. First, we can factor 230496:
\[ 230496 = 2^5 \times 3^1 \times 7^4 \]

4. **Identify the two numbers using HCF:**
Since the HCF is 98, we can express the two numbers in terms of their common factor:
Let the two numbers be \( 98a \) and \( 98b \), where \( a \) and \( b \) are coprime (i.e., their HCF is 1).

5. **Set up the equation using the product:**
\[ 98a \times 98b = 230496 \]
Simplifying this gives:
\[ 9604ab = 230496 \]
Dividing both sides by 9604:
\[ ab = \frac{230496}{9604} = 24 \]

6. **Find pairs of coprime factors of 24:**
The pairs of coprime factors of 24 are:
- (1, 24)
- (3, 8)

7. **Calculate the two numbers:**
For the pair (1, 24):
\[ \text{First number} = 98 \times 1 = 98 \]
\[ \text{Second number} = 98 \times 24 = 2352 \]
The sum:
\[ 98 + 2352 = 2450 \]

For the pair (3, 8):
\[ \text{First number} = 98 \times 3 = 294 \]
\[ \text{Second number} = 98 \times 8 = 784 \]
The sum:
\[ 294 + 784 = 1078 \]

8. **Conclusion:**
The possible sums of the two numbers can be either 2450 or 1078.