The teacher gave the children two multiplication problems with negative integers as homework. Ondra wrote both without parentheses and at home, he thought they were subtraction problems (he didn't notice the dots). He correctly solved the modified problems. One of his results differed from the "correct" one only in sign, and the other was -182 times its reciprocal value. What problems did Ondra get for homework?

(Bednářová)
1. Let's first deal with the first pair of numbers. Let $a, b$ be positive integers. Ondra was supposed to calculate the expression $(-a) \cdot(-b)$, but he calculated $-a-b$. We know that:

$$
-a-b=-a b
$$

which is equivalent to the relation

$$
a+b=a b .
$$

Given that $a, b$ are positive integers, we get the only solution $a=2, b=2$.

2. The second pair of numbers satisfies the relation

$$
\begin{aligned}
& a b=(-182) \cdot \frac{1}{-a-b}, \\
& a b=182 \cdot \frac{1}{a+b} .
\end{aligned}
$$

After rearranging and factoring 182 into prime factors, we get

$$
a b(a+b)=1 \cdot 2 \cdot 7 \cdot 13
$$

Given the symmetry of the numbers $a, b$, we can assume that $a \leq b$ and systematically examine all possibilities of the given factorization:

| $a=1$ | $b=2$ | $a+b=3$ | does not satisfy |
| :--- | :--- | :--- | :--- |
| $a=1$ | $b=7$ | $a+b=8$ | does not satisfy |
| $a=1$ | $b=13$ | $a+b=14=2 \cdot 7$ | SATISFIES |
| $a=1$ | $b=2 \cdot 7$ | $a+b=15$ | does not satisfy |
| $a=2$ | $b=7$ | $a+b=9$ | does not satisfy |
| $a=2$ | $b=13$ | $a+b=15$ | does not satisfy |
| $a=7$ | $b=13$ | $a+b=20$ | does not satisfy |

From the table and the symmetry, it follows that Ondra could have calculated either $(-1) \cdot(-13)$ or $(-13) \cdot(-1)$.

(Complete solution 6 points)