GS. 4 How many positive integers $m$ are there for which the straight line passing through points $A(-m, 0)$ and $B(0,2)$ and also passes through the point $P(7, k)$, where $k$ is a positive integer?
Let the slope of the variable straight line be $a$. Then its equation is: $y=a x+2$ It passes through $A(-m, 0)$ and $P(7, k):\left\{\begin{aligned}-a m+2 & =0 \cdots(1) \\ 7 a+2 & =k \cdots(2)\end{aligned}\right.$ $7(1)+m(2): 14+2 m=k m \Rightarrow m(k-2)=14$ $m=1, k=16$ or $m=2, k=9$ or $m=7, k=4$ or $m=14, k=3$ Number of positive integral values of $m$ is 4 .