Example 2 The minimum 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) $\frac{1}{30}$
(2000, National High School Mathematics Competition)
Explanation: First, use the point-to-line distance formula to get the distance, then conduct the analysis.

Convert the given line to $25 x-15 y+12=0$. Let the distance from the integer point $\left(x_{0}, y_{0}\right)$ on the plane to the line be
$$
d=\frac{|25 x_{0}-15 y_{0}+12|}{5 \sqrt{34}} .
$$

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).