Example 4 Two seventh-grade students are allowed to participate in the chess competition for eighth-grade students. Each pair of players competes once, with a win earning 1 point, a draw earning half a point, and a loss earning zero points. The two seventh-grade students scored a total of 8 points, and each eighth-grade student scored the same number of points. How many eighth-grade students participated in the chess competition? Is the answer unique? 
(8th Canadian Intermediate Mathematics Competition)
Solution: Let there be $n$ eighth-grade students participating, each scoring $k$ points, so the total score of the seventh and eighth-grade students is $8+k n$ points. Since the sum of the scores of both sides in each match is definitely 1 point, it follows that the number of matches played among the participants should equal their total score, i.e., 
$$
8+k n=C_{n+2}^{2},
$$
which simplifies to
$$
n^{2}-(2 k-3) n=14 \text {. }
$$

Thus, $n \mid 14$, and $n$ can be $1, 2, 7$, or 14.
If $n=1$ or 2, then $k<0$, which is impossible;
If $n=7$, then $k=4$;
If $n=14$, then $k=8$.
Therefore, $n$ has two possible values: 7 and 14.