Problem:
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"?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If the slope is 0 , then the line is horizontal and its equation is for some real number . If is an integer, then the line will contain infinitely many lattice points, and if is not an integer, then it will contain no lattice points. Therefore exactly two of the entries in that row of the table are "Possible".
Next suppose that the equation of the line is , where the slope is a nonzero rational number, say for integers and with . If the line contains a lattice point , then it also contains the lattice points , and so on. Therefore the fourth entry in that row of the table is "Possible" and the second and third entries are "Not Possible". To see that the line may contain no lattice points, let be irrational. Then is a point on the line, but if were a lattice point on the line, then
would be an irrational number, a contradiction. Thus the first entry in the "nonzero rational slope" row of the table is "Possible". (This case actually includes the case of zero slope.)
Finally suppose that the equation of the line is , where the slope is an irrational number. The line could certainly contain exactly one lattice point; for example, the equation of the line could\
be and the only lattice point on the line is . It could also contain no lattice points; for example, its equation could be . But if a nonvertical line contains two or more lattice points, say and with , then its slope, , is rational. Therefore the first and second entries in the bottom row of the table are "Possible" and the third and fourth entries are "Not Possible".
In all, of the 12 entries are "Possible" (indicated by in the table below), and 6 are "Not Possible" (indicated by NP).
The problems on this page are the property of the MAA's American Mathematics Competitions