discrete mathematics - Calculate $15^{843} pmod{11}$

Calculate $15^{843} \pmod{11}$

My solution

Fermat's little theorem

Since $15 \equiv 4 \pmod{11}$ and according Fermat's Little Theorem

$$4^{10} \equiv 1 \pmod{11}\;,$$ we shall have

$$15^{843} \equiv 4^{843} \equiv 4^{840} \times 4^3 \equiv (4^{10})^{84} \times 4^3 \equiv 4^3 \equiv 64 \equiv 9 \pmod{11}$$

Is this correct$?$

Answer

Is this correct?

It is.

Fermat's little theorem is indeed one very useful tool to finding the answer, and in case you have any doubt whether you got that part right, you can verify that:

$$4^{10}=1048576=11\times 95325+1\equiv1\pmod{11}$$

The rest is just using the properties of modular arithmetic to replace some terms by congruent but simpler terms. Done explicitely, and correctly, so I see no reason to doubt the result.