Question 6
Let $a_{0}, a_{1}, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with
$$
\sum_{j \in S} a_{j} \equiv i \pmod{2048}.
$$
Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_{j}$ is divisible by $2^{i}$ but not by $2^{i+1}$.
Note: $\sum_{j \in S} a_{j}$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_{j} = a_{2} + a_{5}$, while, for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_{j} = 0$.
$$
\sum_{j \in S} a_{j} \equiv i \pmod{2048}.
$$
Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_{j}$ is divisible by $2^{i}$ but not by $2^{i+1}$.
Note: $\sum_{j \in S} a_{j}$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_{j} = a_{2} + a_{5}$, while, for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_{j} = 0$.