2. Find a true fraction greater than $\frac{1}{3}$ with the following properties:

a) there exists a natural number $x$ such that the value of the fraction does not change if the numerator is increased by $x$ and the denominator is multiplied by $x$,
b) the numerator and the denominator are coprime numbers.

Determine all such fractions.
Solution. Let the fraction be $\frac{a}{b}$, where $b \neq 0$ and $\operatorname{GCD}(a, b)=1$. Then $\frac{a}{b}=\frac{a+x}{b x}$, from which we get $x=\frac{a}{a-1}=1+\frac{1}{a-1}$. Since $x$ is a natural number, we get $a-1$, i.e., $a=2$. According to this, $x=2$. For the fractions

$$
\frac{2}{3}=\frac{2+2}{2 \cdot 3}, \frac{2}{5}=\frac{2+2}{2 \cdot 5}
$$

all conditions of the problem are satisfied. For the fraction $\frac{2}{4}$, the numerator and the denominator are not coprime numbers, and for $b \geq 6$ it holds that $\frac{2}{b} \leq \frac{1}{3}$.