Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.
1. **Understanding the Problem:**
   We need to paint each of the numbers \(1, 2, 3, \ldots, 100\) with one of \(n\) colors such that any two distinct numbers whose sum is divisible by 4 are painted with different colors. We aim to find the smallest value of \(n\).

2. **Analyzing the Conditions:**
   - If \(x + y \equiv 0 \pmod{4}\), then \(x\) and \(y\) must be painted with different colors.
   - The numbers \(1, 2, 3, \ldots, 100\) can be divided into four residue classes modulo 4: \(0, 1, 2, 3\).

3. **Residue Classes:**
   - Numbers \(x \equiv 0 \pmod{4}\): \(4, 8, 12, \ldots, 100\) (25 numbers)
   - Numbers \(x \equiv 1 \pmod{4}\): \(1, 5, 9, \ldots, 97\) (25 numbers)
   - Numbers \(x \equiv 2 \pmod{4}\): \(2, 6, 10, \ldots, 98\) (25 numbers)
   - Numbers \(x \equiv 3 \pmod{4}\): \(3, 7, 11, \ldots, 99\) (25 numbers)

4. **Coloring Constraints:**
   - Numbers in the same residue class \(0 \pmod{4}\) or \(2 \pmod{4}\) must be painted with different colors because their sums are divisible by 4.
   - Numbers in residue classes \(1 \pmod{4}\) and \(3 \pmod{4}\) can be painted with the same color since their sums are not divisible by 4.

5. **Minimum Number of Colors:**
   - Since there are 25 numbers in each of the residue classes \(0 \pmod{4}\) and \(2 \pmod{4}\), and each number in these classes must be painted with a different color, we need at least 25 colors.
   - We can use these 25 colors to paint the numbers in the residue classes \(0 \pmod{4}\) and \(2 \pmod{4}\).
   - For the numbers in the residue classes \(1 \pmod{4}\) and \(3 \pmod{4}\), we can reuse any two of the 25 colors already used.

6. **Constructing the Coloring:**
   - Assign a unique color to each pair of numbers \(4k-2\) and \(4k\) where \(k\) is a positive integer less than or equal to 25.
   - Use one of the 25 colors to paint all numbers \(x \equiv 1 \pmod{4}\).
   - Use another of the 25 colors to paint all numbers \(x \equiv 3 \pmod{4}\).

Conclusion:
The smallest value of \(n\) for which such a situation is possible is \(25\).

The final answer is \(\boxed{25}\).