5. The minimum distance from an integer point (a point with both coordinates as integers) 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. $\frac{1}{30}$
Let $A(m, n)(m, n \in \mathbf{Z})$ be the point and the distance to the line $y=\frac{5}{3} x+\frac{4}{5}$ is
$$
d=\frac{\left|\frac{5}{3} m-n+\frac{4}{5}\right|}{\sqrt{1+\frac{25}{9}}}=\frac{\left|5 m-3 n+\frac{12}{5}\right|}{\sqrt{34}}=\frac{|5(5 m-3 n)+12|}{5 \sqrt{34}} \text {. }
$$

Notice that $5(5 m-3 n)$ is a multiple of 5, then when $m=2, n=4$, $|5(5 m-3 n)+12|_{\min }=2$. Thus, $d_{\min }=\frac{\sqrt{34}}{85}$. Therefore, the answer is $B$.