Divisibility with factorials

Question

Find all positive integers $n$, less than 17, for which $n!+(n+1)!+(n+2)!$ is an integral multiple of 49.

I tried to factor the expression, but I am not having any luck.

Answer 1

$\textbf{Hint:}$ $n!+(n+1)!+(n+2)!=n! \times (n+2)^2$

After that I think you can quite easily find the answers which are:

$5,12,14,15,16$