Sum of the squares of the two consecutive positive integers is 365. The sum of the numbers is :

1. 27

2. 31

3. 25

4. 29

Let two consecutive positive integers be x and x + 1.

Given,

The sum of squares of the two consecutive positive integers = 365.

⇒ x² + (x + 1)² = 365

⇒ x² + x² + 2x + 1 = 365

⇒ 2x² + 2x + 1 - 365 = 0

⇒ 2x² + 2x - 364 = 0

⇒ 2(x² + x - 182) = 0

⇒ x² + x - 182 = 0

⇒ x² + 14x - 13x - 182 = 0

⇒ x(x + 14) - 13(x + 14) = 0

⇒ (x - 13)(x + 14) = 0

⇒ (x - 13) = 0 or (x + 14) = 0     [Using zero -product rule]

⇒ x = 13 or x = -14.

Since, they are consecutive positive integers, x ≠ -14.

⇒ x + 1 = 13 + 1 = 14.

The sum of numbers is = 13 + 14 = 27.

Hence, option 1 is the correct option.