Question

If $5^{97}$ is divided by $52$ then remainder is

• A. $0$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is D $5$

$\because 5^{97}=5.5^{96}=5(25{)}^{48}$
$\Rightarrow 5^{97}=5.(26-1{)}^{48}$
$=5\left\{\right(26{)}^{48}{-}^{48}{C}_1(26{)}^{47}+....{+}^{48}{C}_{46}(26{)}^2{-}^{48}{C}_{27}26+1\}$
$=5\{52\lambda +1)=52\left(m\right)+5\}$
$\Rightarrow$ Remainder is $5$
Hence choice (d) is correct answer.