Q:

How to determine the smallest value of $N=n^4+6n^3+11n^2+6n$ if 13 and 19 both divide N?

I tried to solve for an integer solution by making N equal to multiples of 247 but this is not leading me anywhere. I then tried using the tests for divisibility which did not seem to lead me anywhere either.

A:

It means that $n$ is a natural number and  $N=n^4+6n^3+11n^2+6n.$
I think the fact $$N=n(n+1)(n+2)(n+3)$$ can help.
I see that $n=36$ is valid and check smaller values of $n$.

