$\displaystyle 10^{2001}+1$ is divisible by 11

$\displaystyle \varphi(11)=10\Rightarrow 10^{10}\equiv 1 \ \mbox{(mod 11)}$

$\displaystyle 10^{10*200+1}+1=(10^{10})^{200}*10+1=1^{200}*10+1\ Rightarrow$$\displaystyle 11\equiv 0 \ \mbox{(mod 11)}$

Is this correct?