Question

If $5^{97}$ is divided by $52$, the remainder obtained is

• A. $3$
• B. $5$
• C. $4$
• D. $0$
• E. $1$

Answer 1

The correct option is B $5$

We know that, $5^4=625=52\times 12+1$

$\Rightarrow 5^4=52\lambda +1$, where $\lambda$ is a positive integer.

$\Rightarrow (5^4{)}^{24}=(52\lambda +1{)}^{24}$$={}^{24}{C}_0(52\lambda {)}^{24}+{}^{24}{C}_1(52\lambda {)}^{23}+{}^{24}{C}_2(52\lambda {)}^{22}+--+{}^{24}{C}_0(52\lambda )+{}^{24}{C}_{24}$ (by binomial theorem)

$\Rightarrow 5^{96}=52{[}^{24}{C}_0{52}^{23}{\lambda }^{24}{}^{24}{+}^{24}{C}_1{52}^{23}{\lambda }^{22}+---{-}^{24}{C}_{23}\lambda ]+1$$=\left(\text{a multiple of 52}\right)+1$

On multiplying both sides by $5$, we get

$5^{97}=5^{96}.5=5$ (a multiple of $52$)$+5$

Hence, the required remainder is $5$