Question

What is the remainder when $2^{63}$ is divided by $7$?

• A. $2$
• B. $4$
• C. $1$
• D. $5$

Answer 1

The correct option is C $1$

$2^{63}$ $=2^{3\left(21\right)}$ $=$ $8^{21}$ $=$ ${(7+1)}^{21}$

$=$ ${}^{21}{C}_0$.$7^{21}$ $+$ ${}^{21}{C}_2$.$7^{20}$ $+$ ..........${}^{21}{C}_{20}$.$7^1$ $+$ ${}^{21}{C}_{21}$

On dividing by $7$, we get

$=7$ $($ ${}^{21}{C}_0$.$7^{20}$ $+$ ${}^{21}{C}_2$.$7^{19}$ $+$ ..........$+$ ${}^{21}{C}_{20}$ $)+1$

$=7k+1$

So, $1$ is the remainder