The LCM of 72, 108 and N is 432. If their HCF is 36, then which of the following numbers can be one of the values of N?

Options

A. 64
B. 72
C. 108
D. 144

Answer

To solve the problem, we need to find the value of \( N \) given that the LCM of 72, 108, and \( N \) is 432, and the HCF (or GCD) of these numbers is 36. We will use the relationship between LCM, HCF, and the product of numbers.

### Step-by-Step Solution:

1. **Identify the relationship between LCM, HCF, and the numbers:**
We know that:
\[ \text{LCM}(a, b, c) \times \text{HCF}(a, b, c) = a \times b \times c \]
In our case:
\[ \text{LCM}(72, 108, N) \times \text{HCF}(72, 108, N) = 72 \times 108 \times N \]

2. **Substitute the known values:**
Given:
\[ \text{LCM} = 432 \quad \text{and} \quad \text{HCF} = 36 \]
We can substitute these values into the equation:
\[ 432 \times 36 = 72 \times 108 \times N \]

3. **Calculate the left-hand side:**
First, calculate \( 432 \times 36 \):
\[ 432 \times 36 = 15552 \]

4. **Calculate the product of 72 and 108:**
Next, calculate \( 72 \times 108 \):
\[ 72 \times 108 = 7776 \]

5. **Set up the equation to find \( N \):**
Now we can set up the equation:
\[ 15552 = 7776 \times N \]

6. **Solve for \( N \):**
To find \( N \), divide both sides by 7776:
\[ N = \frac{15552}{7776} \]
Simplifying this:
\[ N = 2 \]

7. **Check if \( N \) can be one of the values:**
Since \( N \) must be a multiple of the HCF (36), we can check:
\[ N = 144 \]

### Conclusion:
The possible value of \( N \) that satisfies the conditions given in the problem is 144.