1. Let $p$ be a prime number. Find all natural numbers $x$ and $y$ that satisfy the equation $p \cdot(x-5)=x \cdot y$.
I/1. Since $p$ is a prime number, we distinguish 2 cases: $p$ divides $x$ or $y$. Let's first consider the case where $p$ divides $x$. Then we can write $x=p x^{\prime}$ (where $x^{\prime}$ is a positive number) and thus get $p x^{\prime}-5=x^{\prime} y$, which we can write as $x^{\prime}(p-y)=5$. Then $x^{\prime}=1$ and $y=p-5$ or $x^{\prime}=5$ and $y=p-1$. However, $x=p$ and $y=p-5$ is a solution only when $p>5$, and $x=5 p$ and $y=p-1$ for any $p$.

Now let's consider the case where $p$ divides $y$. Then $y=p y^{\prime}$ and thus $x-5=x y^{\prime}$ or $x\left(1-y^{\prime}\right)=5$. This equation has no solutions, since $x$ and $y^{\prime}$ are positive numbers.