Key Concepts and Formulas

• Cauchy's Functional Equation (for Natural Numbers): For a function $f: \mathbb{N} \to \mathbb{N}$, the property $f(m+n) = f(m) + f(n)$ for all $m, n \in \mathbb{N}$ implies that $f(k) = k \cdot f(1)$ for any natural number $k$.
• Evaluation of Function Values: Once the general form of the function is known, specific values can be calculated by substituting the input into the general form.

Step-by-Step Solution

Step 1: Determine the General Form of the Function

We are given the functional equation $f(m+n) = f(m) + f(n)$ for all $m, n \in \mathbb{N}$. This is a form of Cauchy's functional equation. For functions defined on natural numbers, this property leads to a linear relationship.

Let's find the form of $f(k)$ for any $k \in \mathbb{N}$. For $k=2$: $f(2) = f(1+1) = f(1) + f(1) = 2f(1)$ For $k=3$: $f(3) = f(2+1) = f(2) + f(1)$ Substituting $f(2) = 2f(1)$: $f(3) = 2f(1) + f(1) = 3f(1)$ By induction, or by observing the pattern, we can conclude that for any natural number $k$: $f(k) = k \cdot f(1)$ This means the function is linear with a slope equal to $f(1)$. Let $f(1) = c$. Then, $f(k) = c \cdot k$. Since the domain and co-domain are $\mathbb{N}$, $c$ must be a positive integer.

Step 2: Use the Given Information to Find the Value of $f(1)$

We are given that $f(6) = 18$. Using the general form derived in Step 1, we can substitute $k=6$: $f(6) = 6 \cdot f(1)$ Now, substitute the given value $f(6) = 18$: $18 = 6 \cdot f(1)$ To find $f(1)$, divide both sides by 6: $f(1) = \frac{18}{6}$ $f(1) = 3$ So, the constant $c$ is 3. The function is $f(k) = 3k$.

Step 3: Calculate $f(2)$ and $f(3)$

Now that we know the specific form of the function, $f(k) = 3k$, we can calculate $f(2)$ and $f(3)$.

For $f(2)$: $f(2) = 3 \cdot 2$ $f(2) = 6$

For $f(3)$: $f(3) = 3 \cdot 3$ $f(3) = 9$

Step 4: Calculate the Product $f(2) \cdot f(3)$

The question asks for the value of $f(2) \cdot f(3)$. Using the values calculated in Step 3: $f(2) \cdot f(3) = 6 \cdot 9$ $f(2) \cdot f(3) = 54$

Common Mistakes & Tips

• Understanding the Functional Equation: The property $f(m+n) = f(m) + f(n)$ is a key indicator of a linear function when the domain is $\mathbb{N}$. Always try to express $f(k)$ in terms of $f(1)$.
• Using Given Data: Do not assume $f(1)$ has a specific value. Always use the given information (like $f(6)=18$) to determine the exact value of $f(1)$.
• Domain and Co-domain: Remember that $f: \mathbb{N} \to \mathbb{N}$. This ensures that $f(1)$ is a positive integer, and consequently, $f(k)$ will also be positive integers for all $k \in \mathbb{N}$.

Summary

The given functional equation $f(m+n) = f(m) + f(n)$ for $f: \mathbb{N} \to \mathbb{N}$ implies that $f(k) = k \cdot f(1)$ for any natural number $k$. Using the provided information $f(6) = 18$, we determined that $f(1) = 3$, thus the function is $f(k) = 3k$. We then calculated $f(2) = 3 \cdot 2 = 6$ and $f(3) = 3 \cdot 3 = 9$. The product $f(2) \cdot f(3)$ is $6 \cdot 9 = 54$.

The final answer is $\boxed{54}$.