The numbers $203$ and $298$ divided with the positive integer $x$ give both remainder $13$. Which are the possible values of $x$ ?
1. Given that the numbers \(203\) and \(298\) divided by the positive integer \(x\) both give a remainder of \(13\), we can write the following congruences:
   \[
   203 \equiv 13 \pmod{x}
   \]
   \[
   298 \equiv 13 \pmod{x}
   \]

2. Subtract \(13\) from both sides of each congruence to simplify:
   \[
   203 - 13 \equiv 0 \pmod{x} \implies 190 \equiv 0 \pmod{x}
   \]
   \[
   298 - 13 \equiv 0 \pmod{x} \implies 285 \equiv 0 \pmod{x}
   \]

3. This means that \(x\) must be a common divisor of \(190\) and \(285\). We need to find the greatest common divisor (GCD) of these two numbers.

4. First, find the prime factorizations of \(190\) and \(285\):
   \[
   190 = 2 \times 5 \times 19
   \]
   \[
   285 = 3 \times 5 \times 19
   \]

5. The common divisors of \(190\) and \(285\) are the products of the common prime factors. The common prime factors are \(5\) and \(19\).

6. Therefore, the possible values of \(x\) are the common divisors of \(190\) and \(285\):
   \[
   x \in \{1, 5, 19, 95\}
   \]

7. However, since \(x\) must be a positive integer greater than \(13\) (as \(13\) is the remainder), we discard \(1\) and \(5\).

8. Thus, the possible values of \(x\) are:
   \[
   x = 19 \quad \text{or} \quad x = 95
   \]

The final answer is \(\boxed{19 \text{ or } 95}\).