Find all integers $n$ greater than or equal to $4$ that satisfy the following conditions:
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[*] Take an arbitrary convex $n$-gon $P$ on the coordinate plane whose vertices are lattice points (points whose coordinates are both integers). There are a total of $n$ triangles that share two sides with $P$. Let $S_1, S_2, \ldots, S_n$ be their areas, and let $S$ be the area of $P$. Then, the greatest common divisor of $2S_1, 2S_2, \ldots, 2S_n$ divides the integer $2S$.
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1. **Restate the problem in mathematical terms:**
   We need to find all integers \( n \geq 4 \) such that for any convex \( n \)-gon \( P \) with vertices at lattice points, the greatest common divisor (gcd) of \( 2S_1, 2S_2, \ldots, 2S_n \) divides \( 2S \), where \( S_i \) are the areas of the triangles sharing two sides with \( P \) and \( S \) is the area of \( P \).

2. **Introduce notation:**
   Let \( A_i = 2S_i \) and \( A = 2S \). We need to show that \(\gcd(A_1, A_2, \ldots, A_n) \mid A\).

3. **Case \( n = 4 \):**
   For a quadrilateral, if the gcd divides \( 2S_1 \) and \( 2S_2 \), it must also divide \( 2(S_1 + S_2) = 2S \). Hence, the condition is satisfied for \( n = 4 \).

4. **General case for \( n \geq 6 \):**
   Consider the construction for \( n = 6 \):
   \[
   (v_1, v_2, v_3, v_4, v_5, v_6) = ((2,0), (0,1), (-2,2), (-1,0), (0,-2), (1,-1))
   \]
   Here, \( 2 \) divides all \( A_i \) but \( A \) is odd. This construction can be extended to \( n > 6 \) by adding vectors that preserve the property \( 2 \mid A_i \) but \( 2 \nmid A \). This violates the condition, so \( n \geq 6 \) does not work.

5. **Case \( n = 5 \):**
   Assume \( p^k \) divides \(\gcd(A_1, A_2, \ldots, A_5)\). We need to show \( p^k \mid A \).

   - Let \( v_1, v_2, v_3, v_4, v_5 \) be the vectors determining the sides of the pentagon.
   - Assume \( v_2 \) has the minimal \( \nu_p(v_2) = a_2 \), and \( \nu_p(v_1) = a_1 \geq \nu_p(v_3) = a_3 \).

6. **Lemma for vector cross products:**
   For two vectors \( v \) and \( w \), \( p^k \mid v \times w \) if:
   - Both coordinates of \( v \) and \( w \) are nonzero modulo \( p \) and \( v \sim w \pmod{p^k} \).
   - \( \nu_p(v_x) = \nu_p(w_x) = j > 0 = \nu_p(v_y) = \nu_p(w_y) \) and \( v' \sim w' \pmod{p^{k-j}} \).

7. **Application of the lemma:**
   - If \( \bar{v_1} \sim \bar{v_2} \pmod{p^{k-a_1-a_2}} \) and \( \bar{v_2} \sim \bar{v_3} \pmod{p^{k-a_2-a_3}} \), then \( \bar{v_1} \sim \bar{v_3} \pmod{p^{k-a_2-a_3}} \).
   - This implies \( \nu_p(v_1 \times v_3) \geq k + a_1 - a_2 \geq k \).

8. **Conclusion for \( n = 5 \):**
   Since \( A = S_2 + S_4 + v_1 \times (v_2 + v_3) \), and \( p^k \mid A \), it follows that \(\gcd(A_1, A_2, \ldots, A_5) \mid A\).

The final answer is \( \boxed{ n = 4 } \) or \( n = 5 \).