2. For two natural numbers, the following property holds: the ratio between their sum, difference, and product is $5: 1: 18$. What are these numbers?
First method:

Let $x$ and $y$ be the required numbers.

According to the condition of the problem, we have: $(x+y):(x-y):(x \cdot y)=5: 1: 18$.

From the extended ratio, it follows that $(x+y):(x-y):(x \cdot y)=(5k):(1k):(18k)$, where $k$ is a natural number, that is,

$x+y=5k, x-y=k, x \cdot y=18k$.

1 POINT

From the system of the first two equations, it follows that $x=3k, y=2k$. 2 POINTS

From the third equation, it follows that $6k^2=18k$, that is, $6k=18$, i.e., $k=3$.

2 POINTS

These are the numbers $x=3 \cdot 3=9$ and $y=2 \cdot 3=6$.

1 POINT

.TOTAL 6 POINTS