5. The minimum distance from 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) $\frac{1}{30}$
(B)
5.【Analysis and Solution】According to the problem, let's assume the integer point is $\left(x_{0}, y_{0}\right)$. It is known that the distance from this point to the line $25 x-15 y+12=0$ is $d=\frac{\left|25 x_{0}-15 y_{0}+12\right|}{\sqrt{25^{2}+(-15)^{2}}}=\frac{\left|25 x_{0}-15 y_{0}+12\right|}{5 \sqrt{34}}$. Since $x_{0}, y_{0} \in \mathbf{Z}$, $25 x_{0}-15 y_{0}$ is a multiple of 5, $\therefore\left|25 x_{0}-15 y_{0}+12\right| \geqslant 2$, and only when $x_{0}=-1, y_{0}=-1$, $\left|25 x_{0}-15 y_{0}+12\right|=2$. Therefore, the minimum value sought is $\frac{\sqrt{34}}{85}$.