24. There is a set of straight lines in the plane such that each line intersects exactly ten others. Which of the following could not be the number of lines in that set?
A 11
B 12
C 15
D 16
E 20
Solution
D

It is convenient in this question to regard a line as being parallel to itself.
Suppose that there are $l$ lines in the given set. Each line intersects all the lines in the set except those that are parallel to it. Therefore, as each line in the given set intersects 10 lines, each line in the set is parallel to $l-10$ lines. We let $k=l-10$.

Then $l=n k$, for some positive integer $n$. The $n k$ lines in the set form $n$ subsets each containing $k$ parallel lines, and such that lines in different subsets are not parallel.
Each line intersects all but $k$ of these $n k$ lines. So each line intersects $n k-k$, that is, $(n-1) k$ lines.
We are given that $(n-1) k=10$. Therefore $n-1$ and $k$ are positive integers whose product is 10. All the possible combinations of values of $n-1$ and $k$ with product 10 are shown in the table below. The third and fourth columns give the corresponding values of $n$ and $n k$, the latter number being the total number of lines in the set.
We see from this table that the only possibilities for the number of lines in a set of lines in which each line intersects 10 other lines are 11,12, 15 and 20. In particular, 16 could not be the number of lines in the set.