Question 7
Let $$
\begin{aligned} N &= \sum_{k=1}^{1000} k \\ &\quad {}\cdot \left(\lceil \log_{\sqrt{2}} k \rceil - \lfloor \log_{\sqrt{2}} k \rfloor\right). \end{aligned}
$$ Find the remainder when $N$ is divided by $1000.$ (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x,$ and $\lceil x \rceil$ denotes the least integer that is greater than or equal to $x.$)
\begin{aligned} N &= \sum_{k=1}^{1000} k \\ &\quad {}\cdot \left(\lceil \log_{\sqrt{2}} k \rceil - \lfloor \log_{\sqrt{2}} k \rfloor\right). \end{aligned}
$$ Find the remainder when $N$ is divided by $1000.$ (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x,$ and $\lceil x \rceil$ denotes the least integer that is greater than or equal to $x.$)