1. Find the largest positive integer $n$ such that $1+2+3+\cdots+n^{2}$ is divisible by $1+2+$ $3+\cdots+n$.
Solution:
1
The statement is
$$
\frac{n(n+1)}{2}\left|\frac{n^{2}\left(n^{2}+1\right)}{2} \Leftrightarrow n+1\right| n\left(n^{2}+1\right)=n^{3}+n .
$$

But $n+1$ also divides $(n+1)\left(n^{2}-n+2\right)=n^{3}+n+2$, so $n+1$ must divide 2. Hence, $n$ cannot be greater than 1 . And $n=1$ clearly works, so that is the answer.