Question

What is the remainder when $5^{68}$ is divided by $8$?

• A. $5$
• B. $1$
• C. $4$
• D. $3$

Answer 1

The correct option is B $1$

$5^{68}$ $=$ ${25}^{34}$ $=$ ${(24+1)}^{34}$

$=$ ${}^{34}{C}_0$${24}^{34}$ $+$ ${}^{34}{C}_1$.${24}^{33}$ $+$ ........$+$ ${}^{34}{C}_{33}$.${24}^1$ $+$ ${}^{34}{C}_{34}$

$=24$ ${}^{34}{C}_0$${24}^{33}$ $+$ ${}^{34}{C}_1$.${24}^{32}$ $+$ ........ + ${}^{34}{C}_{33}$ $)+1$

$=24k+1$

$=8K+1$ h

Hence, the remainder is $1$.