8.96 Find all sequences of natural numbers $a_{1}, a_{2}, a_{3}, \cdots$ that satisfy the following two conditions:
(1) $a_{n} \leqslant n \sqrt{n}, n=1,2,3, \cdots$
(2) For any different $m, n$,
$$m-n \mid a_{m}-a_{n}$$
［Solution］ Let $\left\{a_{n}\right\}$ satisfy the conditions given in the problem, then
$$a_{1}=1, a_{2}=1 \text { or } 2 .$$

If $a_{2}=1$, by (2) we know that for any $n \geqslant 3$, there exist integers $x_{n}$ and $y_{n}$ such that
$$\left\{\begin{array}{l}
a_{n}=1+x_{n} \cdot(n-1) \\
a_{n}=1+y_{n} \cdot(n-2)
\end{array}\right.$$

Since $a_{n}$ is a natural number, $x_{n}, y_{n}$ are non-negative integers and $y_{n} \geqslant x_{n}$. Further, we can get
$$y_{n}=\left(y_{n}-x_{n}\right)(n-1)$$

When $n \geqslant 5$, it is easy to see that $1+(n-1)(n-2)>n \sqrt{n}$, so from (1) and (1) we get $y_{n}-x_{n}=0$, thus
$$x_{n}=y_{n}=0 \text {, for any } n \geqslant 5 \text {, }$$

which means that when $n \geqslant 5$, $a_{n}=1$. Also, by (2) when $n \geqslant 5$
$$n-3\left|a_{3}-1, n-4\right| a_{4}-1$$

Therefore, $a_{3}=a_{4}=1$. In summary, if $a_{1}=1, a_{2}=1$, then
$$a_{n}=1 \text {, for any } n \geqslant 1 \text {. }$$

If $a_{2}=2$. By (2) we know that for $n \geqslant 3$, there exist integers $x_{n}$ and $z_{n}$ such that
$$\left\{\begin{array}{l}
a_{n}=1+x_{n} \cdot(n-1), \\
a_{n}=2+z_{n} \cdot(n-2)
\end{array}\right.$$

From $a_{n}$ being a natural number, we can see that when $n \geqslant 4$, $x_{n}, z_{n}$ are non-negative and
$$z_{n}=1+\left(z_{n}-x_{n}\right)(n-1)$$

Since $z_{n} \geqslant 0$ when $n \geqslant 4$, from (2) we get $z_{n}-x_{n} \geqslant 0$. If $z_{n}-x_{n} \geqslant 1$, then $z_{n} \geqslant n$. Also, since $n \geqslant 4$, $n^{2}-2 n+2>n \sqrt{n}$, so from (1) we get $z_{n}-x_{n}=0$. Thus,
$$x_{n}=z_{n}=1 \text {, for any } n \geqslant 4 \text {, }$$

which means that when $n \geqslant 4$, $a_{n}=n$. Again by (2), we get $10-3 \mid a_{10}-a_{3}$, i.e., $7 \mid 10-a_{3}$. Since $a_{3}$ is a natural number and $a_{3} \leqslant 3 \sqrt{3}$, we have $a_{3}=3$. In summary, if $a_{1}=1, a_{2}=2$, then
$$a_{n}=n, \text { for any } n \geqslant 1 .$$

On the other hand, it is clear that $a_{n}=1$ for any $n \geqslant 1$ and $a_{n}=n$ for any $n \geqslant 1$ both satisfy (1) and (2), so the above two sequences are the required sequences.