Question

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

• A. $3$
• B. $7$
• C. $9$
• D. $11$

Answer 1

Answer: (C)

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