For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers. 
To solve this problem, we need to ensure that for any rectangle with sides parallel to the coordinate axes, the sum of the numbers inside the rectangle is not a prime number. We will consider two interpretations of the problem: one where the boundary is included and one where it is not.

### Case 1: Boundary is not included
1. **Assume all points have the number 4:**
   - If we place the number 4 at every lattice point, then for any rectangle, the sum of the numbers inside the rectangle is a multiple of 4.
   - Since 4 is not a prime number, any multiple of 4 is also not a prime number.
   - Therefore, placing the number 4 at every lattice point satisfies the condition.

2. **Check for smaller numbers:**
   - If we place the number 1 at every lattice point, the sum of the numbers inside any rectangle is simply the number of points inside the rectangle.
   - For a rectangle of size \(m \times n\), the sum is \(m \times n\). If \(m\) and \(n\) are both greater than 1, \(m \times n\) is not a prime number.
   - However, if \(m\) or \(n\) is 1, then the sum is \(m\) or \(n\), which could be a prime number (e.g., 2, 3, 5, etc.).

3. **Conclusion for boundary not included:**
   - The minimal possible value of the maximum number is 4, as placing 4 at every point ensures the sum is always a multiple of 4, which is not a prime number.

### Case 2: Boundary is included
1. **Assume all points have the number 1:**
   - If we place the number 1 at every lattice point, the sum of the numbers inside any rectangle is simply the number of points inside the rectangle.
   - For a rectangle of size \(m \times n\), the sum is \(m \times n\). If \(m\) and \(n\) are both greater than 1, \(m \times n\) is not a prime number.
   - If \(m\) or \(n\) is 1, then the sum is \(m\) or \(n\), which could be a prime number (e.g., 2, 3, 5, etc.).

2. **Assume all points have the number 4:**
   - If we place the number 4 at every lattice point, then for any rectangle, the sum of the numbers inside the rectangle is a multiple of 4.
   - Since 4 is not a prime number, any multiple of 4 is also not a prime number.
   - Therefore, placing the number 4 at every lattice point satisfies the condition.

3. **Conclusion for boundary included:**
   - The minimal possible value of the maximum number is 4, as placing 4 at every point ensures the sum is always a multiple of 4, which is not a prime number.

The final answer is \(\boxed{4}\).