Example 8 Find the smallest positive integer $n$, such that there exist integers $x_{1}, x_{2}, \cdots, x_{n}$, satisfying
$$x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4}=1599$$
From property 1, for any integer $a$, we know that
$$a^{2} \equiv 0(\bmod 4) \text { or } a^{2} \equiv 1(\bmod 8),$$

From this, we can deduce that
$$a^{4} \equiv 0 \text { or } 1(\bmod 16).$$

Using this conclusion, if $n<15$, let
$$x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4} \equiv m(\bmod 16),$$

then
and
$$\begin{array}{c}
m \leqslant n<15 \\
1599 \equiv 15(\bmod 16)
\end{array}$$

This is a contradiction, so
$$n \geqslant 15.$$

Furthermore, when $n=15$, it is required that
$$x_{1}^{4} \equiv x_{2}^{4} \equiv \cdots \equiv x_{n}^{4} \equiv 1(\bmod 16)$$

This means that $x_{1}, x_{2}, \cdots, x_{n}$ must all be odd numbers, which points us in the right direction to find suitable numbers. In fact, among $x_{1}, x_{2}, \cdots, x_{15}$, if one number is 5, twelve are 3, and the other two are 1, then
$$\begin{aligned}
& x_{1}^{4}+x_{2}^{4}+\cdots+x_{15}^{4} \\
= & 5^{4}+12 \times 3^{4}+2 \\
= & 625+972+2 \\
= & 1599
\end{aligned}$$

Therefore, the minimum value of $n$ is 15.