Question

The sum of the squares of two consecutive positive integers is 365. Find the integers.

Answer 1

Let the required two consecutive positive integers be x and (x + 1).

According to the given condition,  

x² + (x + 1)² = 365 

⇒ x² + x² + 2x + 1 = 365 

⇒ 2x² + 2x – 364 = 0 

⇒ x² + x – 182 = 0 

⇒ x² + 14x – 13x – 182 = 0 

⇒ x(x + 14) – 13(x + 14) = 0 

⇒ (x + 14)(x – 13) = 0 

⇒ x + 14 = 0 or x – 13 = 0

∴ x = 13   ...(x is a positive integers) 

When x = 13, 

x + 1 = 13 + 1

= 14 

Hence, the required positive integers are 13 and 14.