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

Let the two consecutive positive integers be x and x+1
Then by 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 or 13
Since the numbers are positive so x = -14 not possible.
So, the numbers are 13 and 13+1 = 14.