1. Find all pairs of natural numbers $a$ and $b$, such that their greatest common divisor is equal to both numbers 30 - $a$ and 42 - $b$.

(Patrik Bak)
Solution. Let $d$ be the greatest common divisor of numbers $a$ and $b$. Since $d$ divides both $a$ and $b$ and also $d=30-a=42-b$, $d$ divides both numbers 30 and 42, and thus also their greatest common divisor 6. For each of the possibilities $d \in\{1,2,3,6\}$, we can easily calculate the values of $a=30-d$ and $b=42-d$ and verify whether $d$ is indeed their greatest common divisor $(a, b)$:

| $d$ | $a$ | $b$ | $(a, b)$ |
| :--- | :--- | :--- | :---: |
| $\mathbf{1}$ | $\mathbf{2 9}$ | $\mathbf{4 1}$ | $\mathbf{1}$ |
| 2 | $28=2^{2} \cdot 7$ | $40=2^{3} \cdot 5$ | 4 |
| $\mathbf{3}$ | $\mathbf{2 7}=3^{3}$ | $\mathbf{3 9}=3 \cdot 13$ | $\mathbf{3}$ |
| 6 | $24=2^{3} \cdot 3$ | $36=2^{2} \cdot 3^{2}$ | 12 |

The values in the first and last columns of the table match only for $d=1$ and $d=3$, so the solutions to the problem are the pairs $a=29, b=41$ and the pairs $a=27, b=39$.