3. Given the equation of line $l$ as $y=\frac{5}{3} x+\frac{4}{5}$, the distance from the closest lattice point to $l$ is equal to ( )
A. $\frac{\sqrt{34}}{85}$
B. $\frac{\sqrt{34}}{170}$
C. $\frac{3 \sqrt{34}}{170}$
D. $\frac{2 \sqrt{34}}{85}$
3. According to the formula for the distance from a point to a line, the distance $d$ from the lattice point $P(x, y)$ to the line $l$ is $d = \frac{|5(3 y-5 x)-12|}{5 \sqrt{34}}$.
Since $x, y \in \mathbf{Z}$, the minimum value of $|5(3 y-5 x)-12|$ is 2, so the minimum value of $d$ is $\frac{2}{5 \sqrt{34}}=\frac{\sqrt{34}}{85}$.