2. (2000 National High School Mathematics League Question) The minimum value of the distance from the lattice points (points with integer coordinates) to the line $y=$ $\frac{5}{3} x+\frac{4}{5}$ is ( )
A. $\frac{\sqrt{34}}{170}$
B. $\frac{\sqrt{34}}{85}$
C. $\frac{1}{20}$
D. $\frac{1}{30}$
2. Convert the known line to $25 x-15 y+12=0$. Let the distance from an integer point $\left(x_{0}, y_{0}\right)$ on the plane to the line be
$$
d=\frac{\left|25 x_{0}-15 y_{0}+12\right|}{5 \sqrt{34}} \text {. }
$$

Since $(25,15)=5$, by Bézout's theorem, $25 x_{0}-15 y_{0}$ represents all multiples of 5. When $25 x_{0}-15 y_{0}=-10$, $d$ reaches its minimum value $\frac{2}{5 \sqrt{34}}=\frac{\sqrt{34}}{85}$. Therefore, the answer is B.