Problem 4. Determine the pairs $(a, b)$ of non-zero natural numbers that have the property that $2 \cdot b+1$ divides $3 \cdot a-1$ and $2 \cdot a+1$ divides $3 \cdot b-1$.
Solution. If the pair $(a, b)$ works, then the numbers $x=\frac{3 a-1}{2 b+1}$ and $y=\frac{3 b-1}{2 a+1}$ are natural

$1 p$

We have $x \cdot y<\frac{3 a}{2 b} \cdot \frac{3 b}{2 a}=\frac{9}{4}=2 \frac{1}{2}$ and, since $x$ and $y$ are natural numbers, we deduce that the following are possible

![](https://cdn.mathpix.com/cropped/2024_06_07_af3c0808b1fd4eefa8c9g-3.jpg?height=54&width=1589&top_left_y=1206&top_left_x=279)

![](https://cdn.mathpix.com/cropped/2024_06_07_af3c0808b1fd4eefa8c9g-3.jpg?height=54&width=1529&top_left_y=1250&top_left_x=339)

![](https://cdn.mathpix.com/cropped/2024_06_07_af3c0808b1fd4eefa8c9g-3.jpg?height=51&width=1529&top_left_y=1297&top_left_x=339)

In the case $x=2, y=1$ we get $a=17, b=12 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \mathbf{1 p}$