2. The magician thought of two natural numbers and told Sime their sum, and told Provo their product. Knowing that the product is 2280, Provo was able to guess the numbers only after Sime mentioned that the sum is odd and two-digit. So what numbers did the magician think of?

[4 points]
# Problem 2.

Answer: 40 and 57.

Let the numbers guessed by the magician be denoted as $a$ and $b$. Then, according to the problem, $a \cdot b = 2280$ and the number $a + b$ is a two-digit odd number. Since the product $a \cdot b$ is even, at least one of the numbers must be even. However, if both numbers were even, their sum would also be even, which is incorrect. Therefore, one number is even, and the other is odd.

Let's consider which prime factors can be part of the odd number. For this, we factorize 2280 into prime factors: $2280 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 19$. Thus, the odd number can only include $3, 5, 19$, and each can appear no more than once, otherwise, the product with the other number would not equal 2280. The remaining factors from this set will be multiplied by 8 to form the second number. Since both numbers must be less than 100, the product of the odd factors not included in the odd number must be less than $100 / 8 = 12.5$. Therefore, the odd number must include 19 and either 3, 5, or both 3 and 5. A small trial shows that $19 \cdot 3 \cdot 5$ does not work (then $19 \cdot 3 \cdot 15 > 100$), and $19 \cdot 5$ also does not work (then the numbers would be $19 \cdot 5 = 95$ and $8 \cdot 3 = 24$, and $95 + 24 > 100$). The only remaining option is $19 \cdot 3 = 57$ and $8 \cdot 5 = 40$, which satisfies all conditions, as the number $57 + 40 = 97$ is a two-digit odd number.