Lattice Points on a Line
In the following table, each question mark is to be replaced by
``Possible'' or ``Not Possible'' to indicate whether a nonvertical line
with the given slope can contain the given number of lattice points
(points both of whose coordinates are integers). How many of the
12 entries will be ``Possible''?
\cline{2-5}
\multicolumn{1}{c|}{} | zero | exactly one
| exactly two | more than two
zero slope | ? | ? | ? | ?
nonzero rational slope | ? | ? | ? | ?
irrational slope | ? | ? | ? | ?
\[
\textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 9
\]
Solutions