Example 7 Let $p_{1}, p_{2}, \cdots, p_{n}$ be $n(\geqslant 2)$ pairwise coprime positive integers, and let
$$\pi_{i}=\frac{p_{1} p_{2} \cdots p_{n}}{p_{i}}, i=1,2, \cdots, n$$

Find the largest positive integer $m$ such that the indeterminate equation
$$\pi_{1} x_{1}+\pi_{2} x_{2}+\cdots+\pi_{n} x_{n}=m$$

has no non-negative integer solutions.
Let $M=(n-1) p_{1} p_{2} \cdots p_{n}-\sum_{i=1}^{n} \pi_{i}$, we prove that the required maximum positive integer is
$M$.

In fact, if there exist non-negative integers $x_{1}, x_{2}, \cdots, x_{n}$, such that
$$\pi_{1} x_{1}+\pi_{2} x_{2}+\cdots+\pi_{n} x_{n}=M$$

Taking modulo $p_{i}$ on both sides of (6), we get
$$\pi_{i} x_{i} \equiv M \equiv-\pi_{i} \quad\left(\bmod p_{i}\right)$$

That is, $p_{i} \mid\left(x_{i}+1\right) \pi_{i}, 1 \leqslant i \leqslant n$.
Using the fact that $p_{1}, p_{2}, \cdots, p_{n}$ are pairwise coprime and the definition of $\pi_{i}$, we know $\left(p_{i}, \pi_{i}\right)=1$, so $p_{i} \mid\left(x_{i}+1\right)$. Since $x_{i}+1 \in \mathbf{N}^{*}$, it follows that $x_{i}+1 \geqslant p_{i}$, i.e., $x_{i} \geqslant p_{i}-1$. This implies (by (6))
$$M=\sum_{i=1}^{n} x_{i} \pi_{i} \geqslant \sum_{i=1}^{n}\left(p_{i}-1\right) \pi_{i}=n p_{1} p_{2} \cdots p_{n}-\sum_{i=1}^{n} \pi_{i}$$

which contradicts the definition of $M$.
On the other hand, if $m>M$, we prove that (5) has a non-negative integer solution.
Note that $\left(\pi_{1}, \pi_{2}, \cdots, \pi_{n}\right)=1$, so (5) has an integer solution $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$. Similar to Example 3, since $\left(x_{1} \pm p_{1}, \cdots, x_{i} \pm p_{i}, \cdots, x_{n}+y p_{n}\right)$, where $y$ is some integer, is also a solution to (5), we can assume $0 \leqslant x_{i} \leqslant p_{i}-1, i=1,2, \cdots, n-1$. At this time,
$$M-\pi_{n}$$

i.e., $\left(x_{n}+1\right) \pi_{n}>0$. Therefore, $x_{n}>-1$, so $x_{n} \geqslant 0$. Hence, when $m>M$, (5) has a non-negative integer solution.

In summary, the required maximum positive integer is
$$M=p_{1} p_{2} \cdots p_{n}\left((n-1)-\sum_{i=1}^{n} \frac{1}{p_{i}}\right)$$