A line with positive rational slope passes through the point A(6, 0) and is at a distance of 5 units from B(1, 3). If the slope of the line is m, then the value of 75m is 

Equation of line passing through point A(6, 0) and having slope m is $(y-0)=m(x-6),m>0$

$\text{ or }mx-y-6m=0,m>0------\left(1\right)$

Distance of this line from point B(1,3) is 5.
$\begin{array}{l}\text{ So, }\left|\frac{m-3-6m}{\sqrt{1+m^2}}\right|=5\\ \Rightarrow (3+5m{)}^2=25\left(1+m^2\right)\\ \Rightarrow 9+30m=25\\ \Rightarrow m=\frac{16}{30}=\frac{8}{15}\end{array}$