If 2^2011 is divided by 13 what is the remainder???

Answer 1

look at the remainders when dividing powers of 2 by 13:
n 2^n mod13
0 1 1
1 2 2
2 4 4
3 8 8
4 16 3
5 32 6
6 64 12
7 128 11
8 256 9
9 512 5
10 1024 10
11 2048 7
12 4096 1
...
so, you see that 2^n mod 13 = 2^(n+12) mod 13

2011 = 167*12 + 7
so, 2^2011 mod 13 = 2^7 mod 13 = 11