The largest exponent of 2 which divides the number  $N=2^{2008}+{10}^{2008}$  is

 $$\begin{gathered} N=2^{2008}+2^{2008}.5^{2008} \\ =2^{2008}[1+5^{2008}]=2^{2008}[1+{(1+4)}^{2008}] \\ =2^{2008}\left[\right(1+1){+}^{2008}{C}_1.4{+}^{2008}{C}_2.4^2+\mathrm{...}{+}^{2008}{C}_{2008}.4^{2008}] \\ =2^{2009}[1+2.{}^{2008}{C}_1+\mathrm{...}]=2^{2009}\underset{odd\Rightarrow notdivisibleby2}{\underbrace{[1+2m],m\in N}} \end{gathered}$$
Hence, highest exponent of  $2=2009$.
Hence, (a) is the correct answer