Question

The sum of a natural number and its reciprocal is `65/8`. Find the number.

Answer 1

Let the natural number be x.

According to the given condition,   

`x + 1/x = 65/8` 

⇒ `(x^2 + 1)/x = 65/8` 

⇒ 8x² + 8 = 65x 

⇒ 8x² – 65x + 8 = 0 

⇒ 8x² – 64x – x + 8 = 0 

⇒ 8x(x – 8) – 1(x – 8) = 0 

⇒ (x – 8)(8x – 1) = 0 

⇒ x – 8 = 0 or 8x – 1 = 0 

⇒ x = 8 or x = `1/8` 

∴ x = 8   ...(x is a natural number) 

Hence, the required number is 8.