Task B-2.3. Andrea and Mirela are preparing for a math competition. When their teacher asked them how many problems they solved yesterday, they did not answer directly. He only found out that Andrea solved fewer problems than Mirela, and each of them solved at least one problem. They also said that the product of the number of problems solved by Andrea and the number of problems solved by Mirela, increased by their sum, is equal to 59. How many problems did Andrea solve? How many different answers are there to this question?
## Solution.

Let $x$ be the number of problems solved by Andrea, and $y$ be the number of problems solved by Mirela, where $x<y$. Then the problem reduces to solving the equation

$$
x y + x + y = 59
$$

which can be rewritten as

$$
x y + x + y + 1 = 60
$$

or

$$
(x+1)(y+1) = 60 \text{.}
$$

The prime factorization of 60 is $60 = 2^2 \cdot 3 \cdot 5$,

so the number of its divisors is 12. Since $x < y$, we have 6 ways to write the number 60 as the product of two natural numbers,

$$
1 \cdot 60, 2 \cdot 30, 3 \cdot 20, 4 \cdot 15, 5 \cdot 12, 6 \cdot 10
$$

Since $x \neq 0$, Andrea could have solved 1, 2, 3, 4, or 5 problems, which means there are 5 different answers to this question.

(2 points)