6. Let $n=m k, k \in \mathbf{N}_{+}$, then $m(7+3 k)=2^{2004} \times 5^{2004}$.
Let $7+3 k=2^{u} \times 5^{v}, u, v \in \mathbf{N}$.
Since both sides of the equation are congruent modulo 3, $u, v$ have the same parity, hence $k$ can take $1003^{2}+1002^{2}-2$ values (excluding $(u, v)=$ $(0,0)$ or $(2,0)$).
Therefore, $(m, n)=\left(\frac{10^{2004}}{7+3 k}, m k\right)$, there are $1003^{2}+1002^{2}-2=2010011$ solutions.
6. (2004 Japan Mathematical Olympiad) Find the number of positive integer pairs $(m, n)$ such that
$$
7 m+3 n=10^{2004} \text {, and } m \mid n \text {. }
$$