Question

Lets $S=\left\{\right(a,b\left)\right\}la,bϵZ,0\le a,b\le 18\}$. The number of lines in $R^2$ passing through $(0,0)$ and exactly one other point in $S$ is?

• A. $16$
• B. $22$
• C. $28$
• D. $32$

Answer 1

The correct option is A $16$

Line passing through origin will be of the form $y=mx$ and any general point on that line can be given by $(h,mh)$.

Now if we take the point as $(a,b)$ with restrictions that a and b both are integers between 0 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 $R^2$ is equal to 16. Therefore, there will be 16 such lines