Jan, Dan, Anna, and Hana each have their favorite number. We know that

- if we add Dan's number and three times Jan's number, the result is an odd number,
- if we subtract Hana's number from Anna's number and multiply this result by five, the result is an odd number,
- if we multiply Dan's number by Hana's number and add 17 to the result, the result is an even number. Determine which of their favorite numbers are odd and which are even.

(E. Semerádová)
When solving, we will repeatedly refer to the properties of sums and products of integers with respect to their parity, which we summarize in the following scheme:

$$
\begin{array}{lll}
\text { even }+ \text { even }=\text { even, } & \text { even }+ \text { odd }=\text { odd, } & \text { odd }+ \text { odd }=\text { even, } \\
\text { even } \times \text { even }=\text { even, } & \text { even } \times \text { odd }=\text { even, } & \text { odd } \times \text { odd }=\text { odd. }
\end{array}
$$

From this and the third condition in the problem, we learn that the product of Dan's and Han's numbers is odd, and thus both of these numbers are odd.

From the second condition, it follows that the difference between Anna's and Han's numbers is odd. Since Han's number is odd, Anna's number must be even.

Since Dan's number is odd, it follows from the first condition that three times Jan's number is even, and thus Jan's number is also even.

Dan's and Han's favorite numbers are odd, Anna's and Jan's numbers are even.

Evaluation. 1 point for determining the parity of each number; 2 points for the quality of the commentary.