# Thread: 10^{2001}+1 is divisible by 11

1. ## 10^{2001}+1 is divisible by 11

$\displaystyle 10^{2001}+1$ 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$$\displaystyle 11\equiv 0 \ \mbox{(mod 11)} Is this correct? 2. Originally Posted by dwsmith \displaystyle 10^{2001}+1 is divisible by 11 \displaystyle \varphi(11)=10\Rightarrow 10^{10}\equiv 1 \ \mbox{(mod 11)} \displaystyle 10^{10*200+1}+1=(10^{10})^{200}*10+1=1^{200}*10+1\ Rightarrow$$\displaystyle 11\equiv 0 \ \mbox{(mod 11)}$

Is this correct?
Yes this is correct. Another way you could've gone is to apply the divisibility rule for 11.

3. Correct!

4. $\displaystyle 10^{2001}+1$ is :

$\displaystyle 1...(2001 \\ times \\ zero)+1$

or

$\displaystyle 1...(2001 \\ times \\ zero)...1$

credit to Mr. chiph588@

5. This is how I would do it:

$\displaystyle 10^{2001}+1=(11-1)^{2001}+1$

On expanding the RHS with the binomial theorem, we see that it is indeed divisible by 11.