Question 149
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”?
| | zero | exactly one | exactly two | more than two |
|----------------------|------|-------------|-------------|---------------|
| zero slope | ? | ? | ? | ? |
| nonzero rational slope | ? | ? | ? | ? |
| irrational slope | ? | ? | ? | ? |
| | zero | exactly one | exactly two | more than two |
|----------------------|------|-------------|-------------|---------------|
| zero slope | ? | ? | ? | ? |
| nonzero rational slope | ? | ? | ? | ? |
| irrational slope | ? | ? | ? | ? |