3. There are several table tennis representative teams, and players from different teams all play one match against each other, while players from the same team do not play against each other. The match statistician recorded the results, and this time, there were 10 players in total, and 27 matches were played.
(1) How many representative teams were there in this competition? Why?
(2) How many players are there in each of these teams?
Solution: (1) Let's assume that this competition is divided into two teams, each with $a_{1}, a_{2}$ members.

From the problem, we have $a_{1}+a_{2}=10, a_{1} a_{2}=27$. This system of equations has no integer solutions (indeed, no real solutions). Therefore, dividing into two teams is incorrect.
(2) Let's assume that this competition has three teams, each with $a_{1}, a_{2}, a_{3}\left(a_{1} \leqslant a_{2} \leqslant a_{3}\right)$ members.
Thus, we have
$a_{1}+a_{2}+a_{3}=10$,
$a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{1}=27$.
We can derive $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=46$.
Therefore, $\frac{46}{3} \leqslant a_{3}^{2} \leqslant 46$, which means $4 \leqslant a_{3} \leqslant 6$.
When $a_{3}=6$, $a_{1}+a_{2}=4, a_{1} a_{2}=3$. Solving this, we get $a_{1}=1, a_{3}=3$. Thus, this competition has $two$ table tennis teams, with the number of members being: $6,3,1$.

When $a_{3}=5$ or 4, $(1,4,5),(2,3,5),(3,3,4),(2,4,4)$ are not integer solutions to the system of equations.
(3) Let's assume that this competition has $i$ teams (4 $\leqslant i \leqslant$ $10$), each with $a_{1}, a_{2}, \cdots, a_{i}$ members. According to (2), we have
$$
\begin{array}{l}
a_{1}+a_{2}+\cdots+a_{i}=10, \\
a_{1}^{2}+a_{2}^{2}+\cdots+a_{i}^{2}=46 .
\end{array}
$$

Assuming $\left(1 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{i} \leqslant 6\right)$, we have
$$
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{i}^{2} \leqslant 42 .
$$
(Equality holds only when $a_{1}=a_{2}=1, a_{3}=2, a_{4}=6$)
Therefore, the system of equations has no positive integer solutions.
In conclusion, this competition has three table tennis teams, with the number of members being 6, 3, and 1, respectively.

Combining the two problems, we get the following question.