## Task 2 - 330822

Susann asks Xaver, Yvonne, and Zacharias to each say a natural number. She then tells them the sum of these three numbers. Each of them multiplies the sum with the number they originally said. As a result, Xaver gets 240, Yvonne gets 270, and Zacharias gets 390.

Determine whether the three original numbers can be uniquely determined! If this is the case, provide these numbers!
Let $x, y, z$ be the three originally mentioned numbers and let $s=x+y+z$ (1) be their sum, then

$$
x \cdot s=240 \quad, \quad y \cdot s=270 \quad, \quad z \cdot s=390
$$

By multiplying (1) with $s$ and then applying (2), it follows that

$$
s^{2}=x \cdot s+y \cdot s+z \cdot s \quad ; \quad 240+270+390=900
$$

Since $s$ as the sum of three natural numbers is itself a natural number, this uniquely determines: It follows that $s=30$ and thus from (2) $x=8, y=9, z=13$.