clearly $n_{max}=(2007^{2007}-2007^{2006}+...-1)-2007$.

Pls let me know how you found it? mathuz wrote: clearly $n_{max}=(2007^{2007}-2007^{2006}+...-1)-2007$.

sengupta_arka123 wrote: Pls let me know how you found it? By the principles of Polynomial Division it is clear that there is some polynomial $P(n)$ and some constant $C$ such that $n^{2007}+n^{2006}+\dotsc+n^2+n+1=(n+2007)P(n)-C$. Inserting $n=-2007$ into this equation you get $C=2007^{2007}-2007^{2006}+\dotsc+2007-1$. Hence $n+2007$ is always a divisor of $n^{2007}+n^{2006}+\dotsc+n^2+n+1+C$. Therefore, the number in the problem is an integer iff $n+2007$ also divides $C$. But the largest $n$ such that $n+2007|C$ is just $n=C-2007$ which is exactly the number in mathuz's post.

Coool!!! Thanks