Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realized that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?
1. Define \( g_i \) as the number of goodbyes received by friend \( i \). Without loss of generality, we can assume \( g_1 < g_2 < \cdots < g_{13} \).

2. Let \( k \) be the number of times Fredek says goodbye to 10 friends. Since each time he says goodbye to 10 friends, the total number of goodbyes is \( 10k \). Therefore, we have:
   \[
   10k = \sum_{i=1}^{13} g_i
   \]

3. Since \( g_{13} \) is the maximum number of goodbyes received by any friend, we have \( g_{13} \le k \).

4. To find the minimum \( k \), we need to consider the sum of the smallest possible values of \( g_i \) given that \( g_1 < g_2 < \cdots < g_{13} \) and \( g_{13} \le k \). The smallest possible values for \( g_i \) are \( g_{13} - 12, g_{13} - 11, \ldots, g_{13} - 1, g_{13} \).

5. Therefore, the sum of \( g_i \) can be written as:
   \[
   \sum_{i=1}^{13} g_i \le (g_{13} - 12) + (g_{13} - 11) + \cdots + (g_{13} - 1) + g_{13}
   \]

6. Simplifying the right-hand side, we get:
   \[
   \sum_{i=1}^{13} g_i \le (k - 12) + (k - 11) + \cdots + (k - 1) + k
   \]

7. This sum is an arithmetic series with 13 terms, where the first term is \( k - 12 \) and the last term is \( k \). The sum of an arithmetic series is given by:
   \[
   \sum_{i=1}^{13} g_i = \frac{13}{2} \left( (k - 12) + k \right) = \frac{13}{2} (2k - 12) = 13k - 78
   \]

8. Since \( 10k = \sum_{i=1}^{13} g_i \), we have:
   \[
   10k \le 13k - 78
   \]

9. Rearranging the inequality, we get:
   \[
   0 \le 13k - 78 - 10k
   \]
   \[
   0 \le 3k - 78
   \]
   \[
   26 \le k
   \]

10. Therefore, the smallest possible value of \( k \) is 26. For \( k = 26 \), it is possible to find an example where each friend receives a different number of goodbyes.

The final answer is \( \boxed{26} \).