If the sum of LCM and HCF of two numbers is 396 and the difference between the LCM and HCF is 324 and the 1st number is 72 then find the second number.

Options

A. A)125
B. B)180
C. C)126
D. D)127

Answer

To solve the problem step by step, we will use the information provided about the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers.

### Step 1: Set up the equations
We know from the problem:
1. The sum of LCM and HCF is 396:
\[ \text{LCM} + \text{HCF} = 396 \quad \text{(Equation 1)} \]
2. The difference between LCM and HCF is 324:
\[ \text{LCM} - \text{HCF} = 324 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations
To find the values of LCM and HCF, we can add and subtract these two equations.

**Adding Equation 1 and Equation 2:**
\[ (\text{LCM} + \text{HCF}) + (\text{LCM} - \text{HCF}) = 396 + 324 \]
This simplifies to:
\[ 2 \times \text{LCM} = 720 \]
So, we can find LCM:
\[ \text{LCM} = \frac{720}{2} = 360 \]

**Subtracting Equation 2 from Equation 1:**
\[ (\text{LCM} + \text{HCF}) - (\text{LCM} - \text{HCF}) = 396 - 324 \]
This simplifies to:
\[ 2 \times \text{HCF} = 72 \]
So, we can find HCF:
\[ \text{HCF} = \frac{72}{2} = 36 \]

### Step 3: Use the relationship between LCM, HCF, and the two numbers
We know that:
\[ \text{LCM} \times \text{HCF} = \text{First Number} \times \text{Second Number} \]
Given that the first number is 72, we can plug in the values:
\[ 360 \times 36 = 72 \times \text{Second Number} \]

### Step 4: Calculate the second number
Calculating the left side:
\[ 360 \times 36 = 12960 \]
Now we set up the equation:
\[ 12960 = 72 \times \text{Second Number} \]
To find the second number, divide both sides by 72:
\[ \text{Second Number} = \frac{12960}{72} = 180 \]

### Conclusion
The second number is **180**.