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 slopenonzero rational slopeirrational slopezero???exactly one???exactly two???more than two???
For zero slope, the line must be horizontal.As shown in the figure, a horizontal line can contain either zero or infinitely many lattice points.
A horizontal line has the equation y=a. If a is an integer, the line contains infinitely many lattice points. Otherwise, it contains none.
Let P mean Possible and NP mean Not Possible. zero slopenonzero rational slopeirrational slopezeroP??exactly oneNP??exactly twoNP??more than twoP??
For a nonzero rational slope, the figure shows that both zero and infinitely many lattice points are possible.Suppose the line passes through a lattice point (x0,y0). y−y0=m(x−x0) Since the slope is rational, let m=ab, where a is a positive integer and b is a nonzero integer.
Then y−y0=ab(x−x0)y=ab(x−x0)+y0. If x−x0=ak,k∈Z, then y=ab(ak)+y0=bk+y0. Thus, (x,y)=(x0+ak,y0+bk) is a lattice point for every integer k.
Therefore, a line with nonzero rational slope contains either zero or infinitely many lattice points. zero slopenonzero rational slopeirrational slopezeroPP?exactly oneNPNP?exactly twoNPNP?more than twoPP?
For an irrational slope, consider y=2x. This line contains the lattice point (0,0).
If x is a nonzero integer, then x2 is irrational, so y cannot be an integer.
Therefore, the line contains exactly one lattice point.
Consider y=x2+21. This line contains no lattice points because:
- If x=0, then y=21, which is not an integer. - If x=0 is an integer, then x2+21 is irrational, so y is not an integer.
Thus, zero lattice points are possible for an irrational slope.
Suppose a nonvertical line contains at least two distinct lattice points (x1,y1) and (x2,y2).
Then its slope is m=x2−x1y2−y1. Since the line is nonvertical, x1=x2.
Because all four coordinates are integers, m is rational.
This contradicts the assumption that the slope is irrational. ×
Therefore, a line with irrational slope cannot contain two or more lattice points.
By Hints 3, 4, and 5, the completed table is zero slopenonzero rational slopeirrational slopezeroPPPexactly oneNPNPPexactly twoNPNPNPmore than twoPPNP Therefore, the number of entries marked Possible is 2+2+2=6.
