2. Two numbers were first multiplied, and then the larger number was divided by the smaller one, resulting in equal outcomes. What are these numbers? How many such pairs of numbers exist?
2. Let $a$ and $b$ be the two numbers mentioned in the problem, with $a > b$. According to the problem,

$$
a \times b = a : b
$$

or

$$
a b^{2} - a = 0
$$

Factoring out the common factor, we get

$$
a \left(b^{2} - 1\right) = 0
$$

From this, we can conclude that $b = 1$, and $a$ can be any number.

Students might reason as follows: find a number such that when multiplied and divided by it, the result is the same. This number is one. Then the first number can be any number, because multiplying it by one and dividing it by one will yield the same number.

Answer: $b = 1$, $a$ can be any number. There are infinitely many such pairs.