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

Let the numbers be x and its reciprocal be $\dfrac{1}{x}$.

Given,

Sum of natural number and it's reciprocal = $\dfrac{65}{8}$

⇒ x + $\dfrac{1}{x} = \dfrac{65}{8}$

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

⇒ 8 × (x² + 1) = 65x

⇒ 8x² + 8 - 65x = 0

⇒ 8x² - 64x -x + 8 = 0

⇒ 8x(x - 8) - 1(x - 8) = 0

⇒ (8x - 1)(x - 8)= 0

⇒ (8x - 1) = 0 or (x - 8) = 0     [Using zero-product rule]

⇒ 8x = 1 or x = 8

⇒ x = $\dfrac{1}{8}$ or x = 8.

Hence, the natural number is 8.