Question

The remainder obtained when $7^{100}$ is divided by $13$ is

• A. 3
• B. 7
• C. 9
• D. 11

Answer 1

Answer: (C)

$7^{100}=\left(7^{2}\right)^{50}=\left(49\right)^{50}$
$=\left(52 - 3\right)^{50}$
$={}^{50}C_{0}^{}\left(52\right)^{50}-{}^{50}C_{1}^{}\left(52\right)^{49}\cdot 3+... -{}^{50}C_{49}^{}\left(52\right)\left(3\right)^{49}+{}^{50}C_{50}^{}\left(3^{50}\right)$
$=13K+3^{50}=13K+\left(3^{3}\right)^{16}\cdot 3^{2}$
$=13K+\left(26 + 1\right)^{16}\cdot 9$
$=13K+9\left({}^{16}C_{0}^{} \cdot \left(26\right)^{16} + {}^{16}C_{1}^{} \cdot \left(26\right)^{15} + . . . + {}^{16}C_{15}^{} \cdot 26 + {}^{16}C_{16}^{}\right)$
$=13K+9\left(13 K_{1} + 1\right)$
$=13\left(K + 9 K_{1}\right)+9$