Question

If $y=\sqrt{56+\sqrt{56+\sqrt{56+...+\infty }}}$, then the number of distinct prime factors of $y$ is

• A. One
• B. Two
• C. Three
• D. Four

Answer 1

The correct option is A One

Given: $y=\sqrt{56+\sqrt{56+\sqrt{56+...+\infty }}}$
$\Rightarrow y=\sqrt{56+y}$
$\Rightarrow y^2=56+y$
$\Rightarrow y^2-y-56=0$
$\Rightarrow y^2-8y+7y-56=0$
$\Rightarrow y(y-8)+7(y-8)=0$
$\Rightarrow (y-8)(y+7)=0$
$\Rightarrow y=8$ or $y=-7$ (Not possible)
Thus, $y=8=2^3$
$\therefore$ Prime factor of $y$ is $2$.
Thus, the number of distinct prime factors of $y$ is one.
Hence, the correct answer is option (1).