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.