The LCM of two numbers is 28 times of their HCF. The sum of their LCM and HCF is 1740. If one of the numbers is 240, find the other number.

Options

A. 240
B. 620
C. 540
D. 420

Answer

To solve the problem step by step, we will follow the information given in the question regarding the relationship between the LCM and HCF of the two numbers, as well as the known values.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
- We know that the LCM (Least Common Multiple) of two numbers is 28 times their HCF (Highest Common Factor).
- Let the HCF be denoted as \( x \).
- Therefore, the LCM can be expressed as:
\[ \text{LCM} = 28x \]

2. **Using the Sum of LCM and HCF**:
- The problem states that the sum of the LCM and HCF is 1740.
- This can be expressed as:
\[ x + 28x = 1740 \]
- Simplifying this, we get:
\[ 29x = 1740 \]

3. **Solving for HCF**:
- To find \( x \), we divide both sides by 29:
\[ x = \frac{1740}{29} \]
- Calculating this gives:
\[ x = 60 \]
- Thus, the HCF is \( 60 \).

4. **Finding the LCM**:
- Now, we can find the LCM using the value of \( x \):
\[ \text{LCM} = 28x = 28 \times 60 = 1680 \]

5. **Using the Relationship Between Numbers**:
- We know that the product of the two numbers is equal to the product of their LCM and HCF:
\[ \text{First Number} \times \text{Second Number} = \text{LCM} \times \text{HCF} \]
- Let the first number be \( 240 \) (as given), and the second number be \( y \):
\[ 240 \times y = 1680 \times 60 \]

6. **Calculating the Second Number**:
- Now we can calculate \( y \):
\[ 240y = 1680 \times 60 \]
- First, calculate \( 1680 \times 60 \):
\[ 1680 \times 60 = 100800 \]
- Now, we have:
\[ 240y = 100800 \]
- Dividing both sides by \( 240 \):
\[ y = \frac{100800}{240} \]
- Calculating this gives:
\[ y = 420 \]

7. **Conclusion**:
- Therefore, the other number is \( 420 \).

### Final Answer:
The other number is **420**.