Question 7
It is known that, for all positive integers $k,$ $$
\begin{aligned} &1^2 + 2^2 + 3^2 + \cdots + k^2 \\ &= \frac{k(k+1)(2k+1)}{6}. \end{aligned}
$$ Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \cdots + k^2$ is a multiple of $200.$
\begin{aligned} &1^2 + 2^2 + 3^2 + \cdots + k^2 \\ &= \frac{k(k+1)(2k+1)}{6}. \end{aligned}
$$ Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \cdots + k^2$ is a multiple of $200.$