## Task 1 - 190731

Determine all ordered pairs $(x ; y)$ of natural numbers that satisfy the following conditions:

(1) The second number $y$ is 1 less than three times the first number $x$.

(2) The product of six times the first number and four times the second number is 1680.
Assume for a pair $(x ; y)$ of natural numbers the conditions (1), (2) are satisfied. Then $y=3 x-1$ (1) and $6 x \cdot 4 y=1680$, so $x y=70$ (2).

By systematically examining all possible cases (considering $70=1 \cdot 2 \cdot 5 \cdot 7$), it is recognized that (2) is only satisfied by the following number pairs: $(1 ; 70),(2 ; 35),(5 ; 14),(7 ; 10),(10 ; 7),(14 ; 5),(35 ; 2),(70 ; 1)$.

However, of these number pairs, only $(5 ; 14)$ also satisfies condition (1). Therefore, only the ordered pair $(5 ; 14)$ can meet all the given conditions.

It indeed satisfies these conditions; for it holds that $14=3 \cdot 5-1$ and $6 \cdot 5 \cdot 4 \cdot 14=30 \cdot 56=1680$.