11. Find the number of positive divisors of $\left(2008^{3}+(3 \times 2008 \times 2009)+1\right)^{2}$.
11. Answer: 91
Let $a=2008$. Then
$$
\begin{aligned}
\left(2008^{3}+(3 \times 2008 \times 2009)+1\right)^{2} & =\left(a^{3}+3 a(a+1)+1\right)^{2} \\
& =\left(a^{3}+3 a^{2}+3 a+1\right)^{2} \\
& =(a+1)^{6}=2009^{6}=7^{12} \cdot 41^{6} .
\end{aligned}
$$

Hence the number of positive divisors is $(12+1)(6+1)=91$.