Question

Let $R =(5 \sqrt{5}+11)^{31}= I +f$, where $I$ is an integer and $f$ is the fractional part of $R$, then $R f$ is equal to -

• A. $2^{31}$
• B. $3^{31}$
• C. $2^{62}$
• D. 1

Answer 1

Answer: (C)

Let $R ^{\prime}=(5 \sqrt{5}-11)^{31}$
Now $ R - R ^{\prime}=(5 \sqrt{5}+11)^{31}-(5 \sqrt{5}-11)^{31}$
$\Rightarrow R - R ^{\prime}=$ Integer $\Rightarrow I +f- R ^{\prime}=$ Integer
$\Rightarrow f- R ^{\prime}$ is an Integer but $-1< f- R ^{\prime}<1$
so $f- R ^{\prime}=0 \Rightarrow f= R ^{\prime}$
so $R . f= R \cdot R ^{\prime}=(5 \sqrt{5}+11)^{31}(5 \sqrt{5}-11)^{31}$
$=4^{31}=2^{62}$