The LCM of two numbers is 24 times of HCF. The sum of HCF and LCM is 375. If first number is 45, find 2nd number

Options

A. 120
B. 110
C. 140
D. 80

Answer

To solve the problem step by step, we will use the relationship between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers, along with the information given in the question.

### Step-by-Step Solution:

1. **Define Variables:**
Let HCF = x. According to the problem, LCM = 24 * HCF = 24x.

2. **Set Up the Equation:**
We know that the sum of HCF and LCM is 375. Therefore, we can write the equation:
\[ HCF + LCM = 375 \]
Substituting the values we have:
\[ x + 24x = 375 \]
This simplifies to:
\[ 25x = 375 \]

3. **Solve for HCF:**
Now, divide both sides by 25 to find x:
\[ x = \frac{375}{25} = 15 \]
So, HCF = 15.

4. **Find LCM:**
Now we can find the LCM using the value of HCF:
\[ LCM = 24 * HCF = 24 * 15 = 360 \]

5. **Use the Relationship Between the Numbers:**
The relationship between the two numbers (let's call them A and B) can be expressed as:
\[ A * B = HCF * LCM \]
We know A = 45 (the first number), HCF = 15, and LCM = 360. Substituting these values into the equation gives:
\[ 45 * B = 15 * 360 \]

6. **Calculate B:**
First, calculate the right side:
\[ 15 * 360 = 5400 \]
Now, substitute this back into the equation:
\[ 45 * B = 5400 \]
To find B, divide both sides by 45:
\[ B = \frac{5400}{45} \]
Calculate the division:
\[ B = 120 \]

### Conclusion:
The second number is **120**.