Question

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

Answer 1

Let the required natural number be $$x$$.

Its reciprocal is $$\frac{1}{x}$$.

According to the problem: $$x + \frac{1}{x} = \frac{65}{8}$$

$$\frac{x^2 + 1}{x} = \frac{65}{8}$$

$$8(x^2 + 1) = 65x$$

$$8x^2 - 65x + 8 = 0$$

Factoring: $$8x^2 - 64x - x + 8 = 0$$

$$8x(x - 8) - 1(x - 8) = 0$$

$$(8x - 1)(x - 8) = 0$$

$$x = \frac{1}{8} \quad \text{or} \quad x = 8$$

Since $$x$$ must be a natural number, reject $$x = \frac{1}{8}$$.

Hence, the required natural number is 8.