When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?
In base $10$, the digits of a perfect square must end in $1, 4, 6, 9, 00$, or $25$. However, for $n\geq 10$, $n!$ is divisible by $100$ and so the last two digits of $n!+10$ are $10$. Therefore, $n!+10$ cannot be a perfect square for $n\geq 10$. And you can just check directly that for $1\leq n\leq 10$, only $n=3$ gives a perfect square.