Question

Al dividir el número $2^{1000}$ entre $13,$ el residuo es:

When the number $2^{1000}$ is divided by $13,$ the remainder in the division is:

$1$

$2$

$3$

$7$

$11$

Usa $2^6\equiv-1\pmod{13}$

Use $2^6\equiv-1\pmod{13}$

Escribe $1000=6\cdot166+4$

Write $1000=6\cdot166+4$

Como $2^6=64\equiv-1\pmod{13},$ $\begin{aligned} 2^{1000}&=2^{6\cdot166+4}\\ &\equiv(-1)^{166}2^4\\ &\equiv16\\ &\equiv3\pmod{13}. \end{aligned}$

Por lo tanto, la respuesta correcta es C.

Since $2^6=64\equiv-1\pmod{13},$ $\begin{aligned} 2^{1000}&=2^{6\cdot166+4}\\ &\equiv(-1)^{166}2^4\\ &\equiv16\\ &\equiv3\pmod{13}. \end{aligned}$

Therefore, the correct answer is C.