Example 2. How many sets of positive integer solutions does the equation $x+2 y+3 z=2000$ have?
Solve: This problem is equivalent to $x-1+2(y-1)+3(z-1) = 2000-6$, i.e.,
$$
a+2 b+3 c=1994
$$

Find the number of non-negative integer solutions.
Let $a=6 a_{1}+r_{1}\left(0 \leqslant r_{1} \leqslant 5\right)$,
$$
\begin{array}{l}
b=3 b_{1}+r_{2}\left(0 \leqslant r_{2} \leqslant 2\right), \\
c=2 c_{1}+r_{3}\left(0 \leqslant r_{3} \leqslant 1\right) .
\end{array}
$$

Equation (1) becomes
$$
\begin{array}{l}
6\left(a_{1}+b_{1}+c_{1}\right)+r_{1}+2 r_{2}+3 r_{3}=1994 . \\
\therefore \quad r_{1}+2 r_{2}+3 r_{3} \equiv 1994 \equiv 2(\bmod 6) .
\end{array}
$$

When $r_{3}=0$, $r_{2}=0, r_{1}=2$, we have
$$
\begin{array}{l}
a_{1}+b_{1}+c_{1}=332 . \\
r_{2}=1, r_{1}=0, a_{1}+b_{1}+c_{1}=332 . \\
r_{2}=2, r_{1}=4, a_{1}+b_{1}+c_{1}=331 .
\end{array}
$$
When $r_{3}=1$, $r_{2}=0, r_{1}=5$, we have
$$
\begin{array}{l}
a_{1}+b_{1}+c_{1}=331 . \\
r_{2}=1, r_{1}=3, a_{1}+b_{1}+c_{1}=331 . \\
r_{2}=2, r_{1}=1, a_{1}+b_{1}+c_{1}=331 .
\end{array}
$$
$\therefore$ Equation (1) has a total of $2 C_{334}^{2}+4 C_{333}^{2}=332334$ non-negative integer solutions.
From the above two examples, we can see that when $a_{i}$ is relatively large, the number of solutions to the congruence
(3) $\sum_{i=1}^{n} a_{i} r_{i} \equiv r_{m} \ (\bmod \ a) \left(0 \leqslant r_{m}<a\right)$ is quite large, making the calculation inconvenient.
$$
\text { Note that, } \begin{aligned}
\sum_{i=1}^{n} a_{i} r_{i} & \leqslant \sum_{i=1}^{n}\left[a_{i}\left(\frac{a}{a_{i}}-1\right)\right] \\
= & n a-\sum_{i=1}^{n} a_{i},
\end{aligned}
$$

Thus, (3) can be transformed into
$$
\begin{array}{c}
a_{1} r_{1}+a_{2} r_{2}+\cdots+a_{n} r_{n}=s a+r_{m}(0 \leqslant s \leqslant \\
{\left[\frac{n a-\sum_{i=1}^{n} a_{i}}{a}\right] \text { or } 0 \leqslant s \leqslant\left[\frac{n a-\sum_{i=1}^{n} a_{i}}{a}\right]-1 \text { and } s}
\end{array}
$$
$\in Z$ ).
For each $s$, let (3) have $t_{i}$ solutions. $x_{1}^{\prime}+x_{2}^{\prime}$ non-negative integer solutions.

Therefore, (1) has a total of $\sum_{i \in 1=1}\left[t_{i} C_{\frac{m-1}{a}-i+n-1}^{a}\right]$ non-negative integer solutions.