5. Given $p_{1}, p_{2}, \cdots, p_{25}$ as 25 distinct prime numbers not exceeding 2004, find the largest positive integer $T$, such that any positive integer not greater than $T$ can always be expressed as the sum of distinct positive divisors of $\left(p_{1} p_{2} \cdots p_{25}\right)^{2004}$ (for example, $1, p_{1}, 1+p_{1}^{2}+p_{1} p_{2}+p_{3}$, etc., are all distinct positive divisors of $\left(p_{1} p_{2} \cdots p_{25}\right)^{2004$).
5. When $p_{1}>2$, 2 cannot be expressed as the sum of distinct positive divisors of $\left(p_{1} p_{2} \cdots p_{25}\right)^{2000}$, at this time $T=1$.
Let $p_{1}=2$, we prove the following more general conclusion:
If $p_{1}, p_{2}, \cdots, p_{k}$ are $k$ distinct primes, $p_{i}2$ when, $T=1$; when $p_{1}=2$, $T=$ $\frac{p_{1}^{2005}-1}{p_{1}-1} \cdot \frac{p_{2}^{2005}-1}{p_{2}-1} \cdots \cdots \frac{p_{25}^{2005}-1}{p_{25}-1$.