11. [15] Find the largest positive integer $n$ such that $1!+2!+3!+\cdots+n$ ! is a perfect square. Prove that your answer is correct.
Answer: 3
Solution: Clearly $1!+2!+3!=9$ works. For $n \geq 4$, we have
$$
1!+2!+3!+\cdots+n!\equiv 1!+2!+3!+4!\equiv 3(\bmod 5),
$$
but there are no squares congruent to 3 modulo 5 .