1. Find the number of subsets $B$ of the set $\{1,2, \cdots, 2005\}$, where the sum of the elements of $B$ is congruent to 2006 modulo 2048.
Notice that each $i(0 \leqslant i \leqslant 2047)$ is exactly the sum of elements of a subset of $\left\{1,2,2^{2}, \cdots, 2^{10}\right\}$.

If set $A$ satisfies that for every $i$, the number of subsets whose sum of elements modulo 2048 equals $i$ does not depend on $i$, then the set $A \cup\{a\}$ still has this property, where $a$ is any number different from the elements of $A$.

Since $\left\{1,2,2^{2}, \cdots, 2^{10}\right\} \subset\{1,2, \cdots, 2005\}$, we know that the number of subsets $B$ of $\{1,2, \cdots, 2005\}$ equals the number of subsets whose sum of elements modulo 2048 is $i(0 \leqslant i \leqslant 2047)$, which is $\frac{2^{2005}}{2048}=2^{1994}$.