Example 1 Find the number of positive integers $a$ that satisfy the following condition: there exist non-negative integers $x_{0}$, $x_{1}, \cdots, x_{2009}$, such that
$$a^{x_{0}}=a^{x_{1}}+a^{x_{2}}+\cdots+a^{x_{2009}} .$$
Solution: Clearly, $a$ that satisfies the condition is greater than 1. Taking both sides of (1) modulo $(a-1)$, we get a necessary condition as
$$1^{x_{0}} \equiv 1^{x_{1}}+1^{x_{2}}+\cdots+1^{x_{2009}}(\bmod a-1)$$

This indicates: $(a-1) \mid 2008$, the number of such $a$ is $=d(2008)=8$.
On the other hand, let $a \in \mathbf{N}^{*}$, satisfying $(a-1) \mid 2008$, set $2008=(a-1) \cdot k$, in $x_{1}, x_{2}, \cdots, x_{2009}$ take $a$ to be $0, a-1$ to be $1, \cdots, a-1$ to be $k-1$, and let $x_{0}=k$, then we know that (1) holds.

Therefore, there are 8 positive integers $a$ that satisfy the condition.