# closest distance between a line and a lattice point in the plane

What is the closest distance between the line $y= 4/7 x + 1/5$ and a lattice point in the plane.

Here is my work:

I re-wrote the equation of the given line as $-20x +35y -7=0$ Then Let $I(a, b)$ be the closest lattice point to the given line.

$d= \frac{|-20a+35b-7|}{5\sqrt{65}}$ . Then what?

For integers $a,b$, the expression $-20a+35b$ can realize any value which is an integer multiple of $\gcd(-20,35)$, that is, any integer multiple of $5$.

Of the integer multiples of $5$, the one that is closest to $7$ is $5$ itself, which can be achieved, for example, using $(a,b)=(5,3)$.

It follows that the minimum value of

$$\frac{|-20a+35b-7|}{5\sqrt{65}}\;\;\text{is}\;\;\frac{2}{5\sqrt{65}}$$