Karolína wrote all three-digit numbers formed by the digits 1, 2, and 3, in which no digit was repeated and 2 was in the tens place. Nikola wrote all three-digit numbers formed by the digits 4, 5, and 6, in which no digit was repeated. Kuba chose one number from Karolína and one number from Nikola so that the sum of these two numbers was even.

What was the digit in the units place of the product of the numbers chosen by Kuba? Determine all possibilities.

(L. Hozová)
Karolína wrote the numbers

$$
123, \quad 321 .
$$

Nikola wrote the numbers

$$
456, \quad 465, \quad 546, \quad 564, \quad 645, \quad 654 .
$$

Both of Karolína's numbers are odd. For an even sum, Kuba had to choose an odd number from Nikola. The even sums are given by the following cases:

$$
123+465, \quad 123+645, \quad 321+465, \quad 321+645
$$

In all cases, the digit in the units place of the product is determined by an odd multiple of 5, which can only be 5.

Evaluation. 1 point for Karolína's and Nikola's numbers; 2 points for Kuba's choice of an even sum; 2 points for determining the last digit of the product.

Note. It is not necessary to list the products to reach the correct conclusion. However, for the quartet of possibilities in $(*)$, these products are, in order,

$$
57195, \quad 79335, \quad 149265, \quad 207045 .
$$