2019 AMC 10A Problem 10

Note that every time the bug crosses a vertical or horizontal line, the bug visits one new tile. This means that the number of tiles the bug visits is $1$ (the first tile) plus the number of lines it crosses. The bug never walks over a corner since $10$ and $17$ are relatively prime, so we don’t have to worry about that. The bug crosses $16$ horizontal lines and $9$ vertical lines for a total of $$1 + 16 + 9 = 26$$ tiles. Thus, C is the correct answer.