Question

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

• A. $3$
• B. $7$
• C. $9$
• D. $11$

Answer 1

Answer: (C)

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