Using Fermat's Theorem compute ..

Question

 So here goes the problem :

 Compute:
 2^1234 (mod 11) 
 How would I go about trying to solve this ?

This is what I have tried so far I know that 2^10 (mod 11) is congruent to 1,but that is about it.

Answer 1

Hint: You're in the right direction. Write $2^{1234} = (2^{10})^{123}\cdot2^4$

Answer 2

Hint:

As an immediate consequence of lil' Fermat, for any integer $a$ not divisible by $11$, one has: $$a^n\equiv a^{n\bmod 10}\mod 11.$$