CGMO 2016 Problem 8

Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$ , define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$ For segment $MN$ , define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$ .

Find the smallest real number $\lambda$ , such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$ , satisfying: for any two points $M,N$ on $l$ , $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$