## Task 2 - 340512

Xaver and Yvette report: Each of us has thought of a natural number. We have shared these numbers with each other.

Xaver says: The successor of my number is divisible by the successor of Yvette's number.

Yvette says: The sum of the successor of my number and the successor of Xaver's number is an odd number.

Anette asks for the product of Xaver's number and Yvette's number: It is 36.

Name two numbers that satisfy these statements! Show that there are no other such numbers!

Hint: The successor of a natural number is the number that is one greater. For example, 115 has the successor 116.
All possibilities to write 36 as a product of two natural numbers are

$$
36=1 \cdot 36, \quad 36=2 \cdot 18, \quad 36=3 \cdot 12, \quad 36=4 \cdot 9, \quad 36=6 \cdot 6
$$

The sum of the successors of two numbers is odd if and only if one of these numbers is even and the other is odd. According to this, only the following possibilities remain:

$$
36=1 \cdot 36, \quad 36=3 \cdot 12, \quad 36=4 \cdot 9
$$

The successors of the numbers mentioned here are 2, 37 respectively, 4, 13 respectively, and 5, 10. In the last case, one of these two successors is divisible by the other.

Thus, the statements are true if and only if Xaver's number is 4 and Yvette's number is 9.