How to do a modular exponentiation? How to solve a problem like this:

Let $N = 11$. What is $2^{2652557887263} \pmod N$?

As part of the questions we are given the multiplication table for $\mathbb{Z}_{11}$:

x | 1  2  3  4  5  6  7  8  9  10
---------------------------------
1 | 1  2  3  4  5  6  7  8  9 10
2 | 2  4  6  8 10  1  3  5  7  9
3 | 3  6  9  1  4  7 10  2  5  8
4 | 4  8  1  5  9  2  6 10  3  7
5 | 5 10  4  9  3  8  2  7  1  6
6 | 6  1  7  2  8  3  9  4 10  5
7 | 7  3 10  6  2  9  5  1  8  4
8 | 8  5  2 10  7  4  1  9  6  3
9 | 9  7  5  3  1 10  8  6  4  2
10|10  9  8  7  6  5  4  3  2  1


...as well as $(\mathbb{Z}_{11}^*,\times)$, the group of units of $\mathbb{Z}_{11}$ under multiplication:

exp|  0   1   2   3   4   5   6   7   8   9   10
 ------------------------------------------------
 g | g^0 g^1 g^2 g^3 g^4 g^5 g^6 g^7 g^8 g^9 g^10
 ------------------------------------------------
 1 |  1   1   1   1   1   1   1   1   1   1   1
 2 |  1   2   4   8   5  10   9   7   3   6   1
 3 |  1   3   9   5   4   1   3   9   5   4   1
 4 |  1   4   5   9   3   1   4   5   9   3   1
 5 |  1   5   3   4   9   1   5   3   4   9   1
 6 |  1   6   3   7   9  10   5   8   4   2   1
 7 |  1   7   5   2   3  10   4   6   9   8   1
 . |
 10|  1  10   1  10   1  10   1  10   1  10   1


Overall, I'm having trouble understanding how the numbers in the table are generated, and how they're used to solve the aforementioned question.
The second row of the second table gives you the values of $g^e\pmod{11}$ for $0 \le e \le 10$. Notice that $2^{10}\pmod{11} \equiv 2^{0}\pmod{11}$. This means that $2^{10}$ raised to any power is congruent to $1\pmod{11}$.

With that in mind, it shouldn't be hard to see that $2^{2652557887263}\pmod{11} \equiv 2^3\pmod{11}$. So the answer is $8$.



P.S. I suspect your question may be slightly off-topic for this site, hence the down votes.