Question

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

• A. 3
• B. 7
• C. 9
• D. 11

Answer 1

Answer: (C)

$7^{100}={\left(7^2\right)}^{50}={\left(49\right)}^{50}$
$={\left(52-3\right)}^{50}$
$={}^{50}{C}_0^{}{\left(52\right)}^{50}-{}^{50}{C}_1^{}{\left(52\right)}^{49}\cdot 3+...-{}^{50}{C}_{49}^{}\left(52\right){\left(3\right)}^{49}+{}^{50}{C}_{50}^{}\left(3^{50}\right)$
$=13K+3^{50}=13K+{\left(3^3\right)}^{16}\cdot 3^2$
$=13K+{\left(26+1\right)}^{16}\cdot 9$
$=13K+9\left({}^{16}{C}_0^{}\cdot {\left(26\right)}^{16}+{}^{16}{C}_1^{}\cdot {\left(26\right)}^{15}+...+{}^{16}{C}_{15}^{}\cdot 26+{}^{16}{C}_{16}^{}\right)$
$=13K+9\left(13K_1+1\right)$
$=13\left(K+9K_1\right)+9$