Getting wrong results for multiple power mod 11

Question

I have

$\ 2^{2^{12}} \pmod {11}$

Using Euler's theorem: $\ 2^{2^{12}\pmod {10}} \pmod {11}$.

Now I am computing $\ {2^{12}} \pmod {11} = 4$.

After that, I am putting ${4}$ as power of ${2}$. --> $\ {2^{4}} \pmod {11} = 5$. But that's not the correct answer.

Actually the result should be: $\ 2^{2^{12}} \pmod {11} = 9$.

I don't what I am doing wrong, just begun with the Euler's theorem, so I would appreciate your help.

Regards.

Answer 1

You computed $2^{12}\bmod 11$ instead of $2^{12}\bmod 10$, which is $6$.

Therefore $\:2^{2^{12}}\bmod 11=2^{6}\bmod 11=64\bmod 11=\cdots$

Answer 2

Why did you compute $2^{12}\pmod{11}$ instead of $2^{12}\pmod{10}$? You started out correctly, but then you didn't follow what you had written down.

$$2^{12}=4096\equiv 6\pmod{10}$$ $$2^6=64\equiv9\pmod{11}$$