Let `f:N to N` for which `f(m+n)=f(m)+f(n) forall m,n in N` .If `f(6)=18` then the value of `f(2)*f(3)` is

Answer

To solve the problem, we need to find the value of \( f(2) \times f(3) \) given that \( f: \mathbb{N} \to \mathbb{N} \) satisfies the functional equation \( f(m+n) = f(m) + f(n) \) for all \( m, n \in \mathbb{N} \) and that \( f(6) = 18 \).

### Step-by-Step Solution:

1. **Understanding the Functional Equation**:
The equation \( f(m+n) = f(m) + f(n) \) suggests that \( f \) is a linear function. A common form for such functions is \( f(n) = k \cdot n \) for some constant \( k \).

2. **Using the Given Information**:
We know that \( f(6) = 18 \). If we assume \( f(n) = k \cdot n \), then:
\[ f(6) = k \cdot 6 = 18 \]
From this, we can solve for \( k \):
\[ k = \frac{18}{6} = 3 \]

3. **Finding \( f(2) \) and \( f(3) \)**:
Now that we have \( k = 3 \), we can find \( f(2) \) and \( f(3) \):
\[ f(2) = 3 \cdot 2 = 6 \]
\[ f(3) = 3 \cdot 3 = 9 \]

4. **Calculating \( f(2) \times f(3) \)**:
Finally, we need to calculate \( f(2) \times f(3) \):
\[ f(2) \times f(3) = 6 \times 9 = 54 \]

Thus, the value of \( f(2) \times f(3) \) is \( 54 \).