$\textbf{Problem 4.}$ What is the largest number such that dividing $17$ or $30$ by this number, the same remainder is obtained in both cases
1. We need to find the largest number \( n \) such that when both 17 and 30 are divided by \( n \), the same remainder is obtained. This can be expressed as:
   \[
   17 \mod n = 30 \mod n
   \]
2. This implies that the difference between 30 and 17 must be divisible by \( n \):
   \[
   30 - 17 \equiv 0 \mod n
   \]
3. Simplifying the difference:
   \[
   30 - 17 = 13
   \]
4. Therefore, \( n \) must be a divisor of 13. The divisors of 13 are 1 and 13.
5. To find the largest such \( n \), we choose the largest divisor of 13, which is 13.

Thus, the largest number \( n \) such that dividing 17 or 30 by \( n \) gives the same remainder is \( \boxed{13} \).