Prepare-se para um confronto imperdível! Athletico-PR x Atlético-GO se enfrentam pela 11ª rodada da Série B do Campeonato Brasileiro. O jogo será realizado na Ligga Arena, em Curitiba, com o Athletico-PR buscando se reabilitar após um empate, enquanto o Atlético-GO tentaFind the remainder when $2^{2015}$ is divided by $17$.
Let us find the remainders when powers of 2 are divided by 17.
$2^1 \equiv 2 \pmod{17}$
$2^2 \equiv 4 \pmod{17}$
$2^3 \equiv 8 \pmod{17}$
$2^4 \equiv 16 \equiv -1 \pmod{17}$
$2^5 \equiv -2 \pmod{17}$
$2^6 \equiv -4 \pmod{17}$
$2^7 \equiv -8 \pmod{17}$
$2^8 \equiv -16 \equiv 1 \pmod{17}$
We have found that $2^8 \equiv 1 \pmod{17}$. Therefore, the remainders repeat every 8 powers.
To find the remainder when $2^{2015}$ is divided by 17, we need to find the remainder when 2015 is divided by 8.
$2015 = 8 \cdot 251 + 7$
So, $2015 \equiv 7 \pmod{8}$.
Thus, $2^{2015} = 2^{8 \cdot 251 + 7} = (2^8)^{251} \cdot 2^7 \equiv 1^{251} \cdot 2^7 \pmod{17} \equiv 1 \cdot 2^7 \pmod{17}$
Now we need to find $2^7 \pmod{17}$.
$2^7 = 128$
$128 = 17 \cdot 7 + 9$
So $128 \equiv 9 \pmod{17}$.
Also, $2^7 \equiv -8 \pmod{17} \equiv -8+17 \pmod{17} \equiv 9 \pmod{17}$
Therefore, $2^{2015} \equiv 2^7 \pmod{17} \equiv 9 \pmod{17}$.
The remainder when $2^{2015}$ is divided by 17 is 9.
Final Answer: The final answer is $\boxed{9}$
