## Task 3 - 241213

Determine all real numbers $x$ for which $2 x-3,5 x-14$ and $\frac{2 x-3}{5 x-14}$ are integers.
For a real number $x$, if the numbers

$$
g=2 x-3 \quad ; \quad h=5 x-14 \quad \text { and } \quad k=\frac{2 x-3}{5 x-14}=\frac{g}{h}
$$

are integers, then from (1) and (2),

$$
5 g=10 x-15 \quad ; \quad 2 h=10 x-28
$$

and therefore

$$
5 g-2 h=13
$$

Considering the equation $g=h k$ derived from (3), we get

$$
(5 k-2) h=13
$$

Thus, $5 k-2$ is a divisor of 13, and therefore one of the numbers $1, -1, 13, -13$. However, only 13 has the form $5 k-2$ with an integer $k$, and thus from (4) we further get $h=1$, and from (2) $5 x=h+14=15, x=3$.

Therefore, only $x=3$ can have the required property. Indeed, $2 \cdot 3-3=3, 5 \cdot 3-14=1$, and $\frac{2 \cdot 3-3}{5 \cdot 3-14}=3$ are integers. Thus, only $x=3$ has the required property.

Adapted from $[5]$