2024 AMC 10B Problem 19

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”?

\textbf{(A)} 4 \qquad \textbf{(B)} 5 \qquad \textbf{(C)} 6 \qquad \textbf{(D)} 7 \qquad \textbf{(E)} 9

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.