Question

The sum of a natural number and its reciprocal is $\frac{65}{8}$. Find the number.

Answer 1

Let the natural number be x.

According to the given condition,
$$\begin{gathered} x+\frac{1}{x}=\frac{65}{8} \\ \Rightarrow \frac{x^2+1}{x}=\frac{65}{8} \\ \Rightarrow 8x^2+8=65x \\ \Rightarrow 8x^2-65x+8=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow 8x^2-64x-x+8=0 \\ \Rightarrow 8x\left(x-8\right)-1\left(x-8\right)=0 \\ \Rightarrow \left(x-8\right)\left(8x-1\right)=0 \\ \Rightarrow x-8=0\mathrm{or}8x-1=0 \end{gathered}$$
$\Rightarrow x=8\mathrm{or}x=\frac{1}{8}$
∴ x = 8 (x is a natural number)

Hence, the required number is 8.