Question

Determine o resto da divisão de 728 ^330^48 por 11? . Por favor explique detalhado .

Answer 1

(1) 
$728\equiv 2 \mod 11\\ \\ 728^5\equiv 2^5\mod 11\\ \\ 728^5\equiv 32\mod 11\\ \\ 728^5\equiv -1\mod 11$

(2)
$728 ^{330^{48}}=\left(728^5\right)^{ \frac{330^{48}}{5} }=\left(728^5\right)^{66^{48}\times 5^{47} }\\ \\ \text{Note que }66^{48}\times 5^{47} \text{ es un n\'umero par}$

(3)
$\left(728^{5}\right)^{66^{48}\times5^{47}}\equiv(-1)^{66^{48}\times5^{47}}\mod11\\ \\ \left(728^{5}\right)^{66^{48}\times5^{47}}\equiv1\mod11\\ \\ 728 ^{330^{48}}\equiv1\mod11$

Resposta: o resto é 1