Lets $S=\{(a, b)\} \mid a, b \in Z, 0 \leq a, b, \leq 18\}$. The number of lines in $R^{2}$ passing through $(0,0)$ and exactly one other point in $S$ is

Line passing through origin will be of the form $\mathrm{y}=\mathrm{mx}$ and any general point on that line can be given by $(\mathrm{h}, \mathrm{mh})$. Now if we take the point as $(a, b)$ with restrictions that a and b both are integers between o and 18 , both included, then there will be $18 \times 18=324$ such integral pairs in the complete $18 \times 18$ square. We need to select those pairs, which do not have any factor i.e coprime pairs in the given range, such as $(1,10),(1,11)$, etc. Number of lines will be equal to number of such points in the plane. Number of such pairs on $\mathrm{R}^{2}$ is equal to 16 . Therefore, there will be 16 such lines