**Solution:**

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

Therefore, as per the given question,

*x*^{2} + (*x* + 1)^{2} = 365

⇒ *x*^{2 }+ *x*^{2 }+ 1 + 2*x* = 365

⇒ 2*x*^{2} + 2x – 364 = 0

⇒ *x*^{2 }+ *x *– 182 = 0

⇒ *x*^{2 }+ 14*x* – 13*x* – 182 = 0

⇒ *x*(*x* + 14) -13(*x* + 14) = 0

⇒ (*x* + 14)(*x* – 13) = 0

Thus, either, *x* + 14 = 0 or *x* – 13 = 0,

⇒ *x* = – 14 or *x* = 13

Since the integers are positive, *x* can be 13 only.

∴ *x* + 1 = 13 + 1 = 14

Therefore, two consecutive positive integers will be 13 and 14.