Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.
1. **Understanding the problem:**
   We need to find the largest positive integer \( n \) (where \( n \ge 3 \)) such that there exists a convex \( n \)-gon where the tangent of each interior angle is an integer.

2. **Properties of the tangent function:**
   - The tangent function, \(\tan x\), is strictly increasing on the interval \(\left(0, \frac{\pi}{2}\right)\).
   - The equation \(\tan x = k\) (where \( k \in \mathbb{N} \)) has solutions in the interval \(\left(0, \frac{\pi}{2}\right)\).

3. **Solutions for \(\tan x = k\):**
   - For \( k = 1 \), the solution is \( x_1 = \frac{\pi}{4} \).
   - For \( k \ge 2 \), the solutions \( x_k \) lie in the interval \(\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\).

4. **Negative tangents:**
   - Using the property \(\tan(\pi - \alpha) = -\tan \alpha\), the equations \(\tan x = -k\) (where \( k \in \mathbb{N} \)) have solutions \( x'_k = \pi - x_k \) in the interval \((0, \pi)\).

5. **Range of angles:**
   - The minimum value of \( x_k \) and \( \pi - x_k \) is \( x_1 = \frac{\pi}{4} \).
   - The maximum value of \( x_k \) and \( \pi - x_k \) is \( \pi - x_1 = \frac{3\pi}{4} \).

6. **Sum of interior angles:**
   - For a convex \( n \)-gon, the sum of the interior angles is \((n-2)\pi\).
   - Let \(\beta_m = \pi - \alpha_m\) be the interior angles of the \( n \)-gon. Then, \(\sum_{m=1}^n \beta_m = (n-2)\pi\).

7. **Sum of \(\alpha_m\):**
   - Since \(\beta_m = \pi - \alpha_m\), we have \(\sum_{m=1}^n \alpha_m = 2\pi\).
   - Given \(\tan \beta_m \in \mathbb{Z} \iff \tan \alpha_m \in \mathbb{Z}\), we need to ensure that the sum of the angles \(\alpha_m\) is \(2\pi\).

8. **Inequality for \( n \):**
   - Since each \(\alpha_m \ge \frac{\pi}{4}\), we have:
     \[
     2\pi = \sum_{m=1}^n \alpha_m \ge n \cdot \frac{\pi}{4}
     \]
   - Simplifying this inequality:
     \[
     2\pi \ge n \cdot \frac{\pi}{4} \implies 2 \ge \frac{n}{4} \implies n \le 8
     \]

9. **Conclusion:**
   - The maximum possible value of \( n \) for which the tangents of the angles of the convex \( n \)-gon are integers is \( n_{\text{max}} = 8 \).
   - In this case, all angles \(\beta_m\) have the measure \(\frac{3\pi}{4}\).

The final answer is \( \boxed{ n_{\text{max}} = 8 } \).