3. Given the sum of several positive integers is 2005, find the maximum value of their product.
3. Find $\max \left\{\prod_{i=1}^{n} x_{i}\right.$; $\left.\sum_{i=1}^{n} x_{i}=2005, x_{i} \in \mathbf{N}^{+}\right\}$.

When $x_{i_{0}}=1$, merging 1 into other $x_{i}$, the sum remains unchanged, but the product is larger; when $x_{i_{0}}>4$, splitting $x_{i_{0}}$ into 2 and $x_{i_{0}}-2$, the sum remains unchanged, but the product is larger; when $x_{i_{0}}=4$, splitting $x_{i_{0}}$ into two 2s, the sum and product remain unchanged; when replacing three 2s with two 3s, the sum remains unchanged, but the product increases. Therefore, when $\prod_{i=1}^{n} x_{i}$ is maximized, it must be in the form of $2^{p} 3^{q}$, and $p<3$. Since $2005=667 \times 3+2 \times 2$, we have $\max \prod_{i=1}^{n} x_{i}=2^{2} \times 3^{667}$.