5. Let $p_{1}, p_{2}, \cdots, p_{k}$ be different primes. Find the number of $p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{k}^{\alpha_{k}}$ that satisfy $\alpha_{i} \in \mathbf{N}, p_{1} p_{2} \cdots p_{k}=\alpha_{1} \alpha_{2} \cdots \alpha_{k}$.
5. Obviously, for any $i \in\{1,2, \cdots, k\}$, there exists $s \in\{1,2, \cdots, k\}$, such that $p_{i} \mid \alpha_{s}$. Then
$$
p_{1} p_{2} \cdots p_{k}=\alpha_{1} \alpha_{2} \cdots \alpha_{k} \text {. }
$$

From this, we can construct an ordered $k$-tuple of positive integers $\left(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}\right)$ such that equation (1) holds.

Take the $k$-tuple $(1,1, \cdots, 1)$. Multiply any one of its terms by $p_{1}$, then multiply any one of the terms in the $k$-tuple by $p_{2}$, indicating that there are $k$ possible methods for $p_{2}$. Similarly, for all primes $p_{3}, \cdots, p_{k}$, there are $k$ possible methods. Therefore, there exist $k^{k}$ ordered $k$-tuples $\left(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}\right)$ such that equation (1) holds.

Obviously, all these $k$-tuples determine some numbers of the form $p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{k}^{\alpha_{k}}\left(\alpha_{i} \in \mathbf{N}\right)$, and different $k$-tuples determine different numbers.
In fact, if
$$
\left(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}\right) \neq\left(\beta_{1}, \beta_{2}, \cdots, \beta_{k}\right),
$$

then there exists at least one $i \in\{1,2, \cdots, k\}$ such that
$$
\alpha_{i} \neq \beta_{i}(i \in\{1,2, \cdots, k\}) .
$$

Thus, $p_{i}^{\alpha_{i}} \neq p_{i}^{\beta_{i}}$, i.e.,
$$
p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{k}^{\alpha_{k}} \neq p_{1}^{\beta_{1}} p_{2}^{\beta_{2}} \cdots p_{k}^{\beta_{k}} \text {. }
$$

Conversely, if $p_{1}^{\alpha_{1}} p^{\alpha_{2}} \cdots p_{k}^{\alpha_{k}}\left(\alpha_{i} \in \mathbf{N}\right)$ satisfies
$$
\alpha_{1} \alpha_{2} \cdots \alpha_{k}=p_{1} p_{2} \cdots p_{k},
$$

then it determines an ordered $k$-tuple of positive integers $\left(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}\right)$ such that equation (1) holds.
In summary, the number of positive integers sought is $k^{k}$.