## Task 1 - 190931

When solving an equation of the form $a x - 6 = b x - 4$ with given natural numbers $a$ and $b$, Matthias notices:

(1) The equation has a natural number $x$ as a solution.

(2) The same number results when, for verification, the found solution $x$ is substituted on each side of this equation.

Determine all pairs $(a, b)$ of natural numbers for which these observations (1) and (2) are true!
The equation is equivalent to $(a-b) x=2$. Furthermore, according to (1) $x \in \mathbb{N}$, so $x \geq 0$, which means $a-b \geq 0$ and thus $a-b \in \mathbb{N}$. In particular, $x$ is a divisor of 2, i.e., $x=1$ or $x=2$.

1. Case: $x=1$. Then, according to (2), $a-6=b-4=1$, so $a=7$ and $b=5$. The check confirms that the pair $(a, b)=(7,5)$ satisfies both properties.
2. Case: $x=2$. Then, according to (2), $2 a-6=2 b-4=2$, so $a=4$ and $b=3$, which is also confirmed by the check. The second pair $(a, b)$ that satisfies the problem statement is thus $(4,3)$, and there are no others.