1. How many four-digit numbers exist that are divisible by 17 and end in 17?
1. Let's denote the four-digit number as $\overline{x y 17}$. Then the number $\overline{x y 17}-17$ is also divisible by 17. But $\overline{x y 17}-17=100 \cdot \overline{x y}+17-17=100 \cdot \overline{x y}$. Since the numbers 100 and 17 are coprime, the two-digit number $\overline{x y}$ is divisible by 17. By enumeration, we find all two-digit numbers divisible by 17: 17, 34, 51, 68, 85. There are five of them. Therefore, there will also be five such numbers: 1717, 3417, 5117, 6817, 8517.

Answer: 5.