Question

Let $R=(5\sqrt{5}+11{)}^{31}=I+f$, where $I$ is an integer and $f$ is the fractional part of $R$, then $Rf$ 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 $R.f=R\cdot R^{\prime }=(5\sqrt{5}+11{)}^{31}(5\sqrt{5}-11{)}^{31}$
$=4^{31}=2^{62}$