If L, M and N are natural number, then value of `LM=M+N=` ?
(A) `L+N=8+M`
(B) `M^(2)=(N^(2))/(L+1)`
(C) `M=L+2`

Options

A. Either A and B or B and C are sufficient to answer the question
B. Either A and B or A and C are sufficient to answer the question
C. Either A and C or B and C are sufficient to answer the question
D. Either only A or B and C together are sufficient to answer the question

Explanation

To solve the problem, we need to analyze the given statements and determine if they provide sufficient information to find the value of \( LM = M + N \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation \( LM = M + N \). This means that the product of \( L \) and \( M \) is equal to the sum of \( M \) and \( N \).

2. **Analyzing Statement (A)**:
The first statement is \( L + N = 8 + M \).
- Rearranging gives us \( N = 8 + M - L \).
- This equation has three variables \( L, M, N \) but only one equation, which is insufficient to solve for a unique value of \( LM \).

**Conclusion**: Statement (A) alone is not sufficient.

3. **Analyzing Statement (B)**:
The second statement is \( M^2 = \frac{N^2}{L + 1} \).
- From \( LM = M + N \), we can express \( N \) in terms of \( L \) and \( M \): \( N = LM - M \).
- Substituting \( N \) into statement (B) gives us:
\[ M^2 = \frac{(LM - M)^2}{L + 1} \]
- This leads to a more complex equation involving \( L \) and \( M \), but we still have two variables and only one equation.

**Conclusion**: Statement (B) alone is also not sufficient.

4. **Analyzing Statement (C)**:
The third statement is \( M = L + 2 \).
- Substituting \( M \) into the original equation \( LM = M + N \) gives:
\[ L(L + 2) = (L + 2) + N \]
- This simplifies to:
\[ L^2 + 2L = L + 2 + N \implies L^2 + L - 2 = N \]
- Here, we have \( N \) expressed in terms of \( L \).

5. **Combining Statements (B) and (C)**:
Now, using both statements (B) and (C):
- From statement (C), substitute \( M = L + 2 \) into statement (B):
\[ (L + 2)^2 = \frac{N^2}{L + 1} \]
- We can now express \( N \) from statement (C) and substitute it back into the equation to solve for \( L \).

6. **Finding Values**:
- Solving the equations from statements (B) and (C) will yield specific values for \( L, M, \) and \( N \).
- After solving, we find \( L = 3 \), \( M = 5 \), and \( N = 10 \).

7. **Final Calculation**:
- Now substituting back, we find:
\[ LM = 3 \times 5 = 15 \]
- And \( M + N = 5 + 10 = 15 \).

### Conclusion:
The value of \( LM = M + N \) is \( 15 \).