**Solution:**

Here, sum of the numbers is 27.

Let one of the numbers be x.

∴ Other number = 27 – x

According to the condition,

Product of the numbers = 182

⇒ x (27 – x) = 182

⇒ 27x – x^{2} = 182

⇒ –x^{2} + 27x – 182 = 0

⇒ x^{2} – 27x + 182 = 0

⇒ x^{2} – 13x – 14x + 182 = 0 –27 = (–13) + (– 14) and

⇒ x (x – 13) – 14 (x – 13) = 0 (– 13) × (– 14) = 182

⇒ (x – 13) (x – 14) = 0

Either x – 13 = 0 ⇒ x = 13

or x – 14 = 0 ⇒ x = 14

Thus, the required numbers are 13 and 14.