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) $\frac{1}{30}$
5. (B).

Let the integer point be $\left(x_{0}, y_{0}\right)$, then its distance 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 $x_{0} y_{0} \in \mathbf{Z}$, hence $25 x_{0}-15 y_{0}$ is a multiple of 5.
Thus, $\left|25 x_{0}-15 y_{0}+12\right| \geqslant 2$.
When $x_{0}=-1, y_{0}=-1$,
$$
\left|25 x_{0}-15 y_{0}+12\right|=2 \text {. }
$$

Therefore, the minimum value sought is $\frac{\sqrt{34}}{85}$.