Question

 Find two consecutive positive integers, sum of whose squares is 365.

Answer 1

Product of two consecutive positive integers = 306

Let first positive integer = x

Second positive integer = x + 1

Sum of squares of both consecutive numbers = 365

(x)² + (x + 1)² = 365

x² + x² + 2x + 1 = 365

2x² + 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 + 14)(x - 13) = 0

x + 14 = 0 and x - 13 = 0

x = -14 and x = 13

Since

First positive integer = x = 13

Second positive integer = x + 1 = 13 + 1 = 14

Thus, the required consecutive positive integers are 13 and 14.