\section*{Problem 1 - 211041}

Determine all pairs \((a ; b)\) of positive integers \(a, b\) that have the property that exactly three of the following four statements (1), (2), (3), (4) are true and one is false! The statements are:

\[
\begin{array}{llll}
b \mid(a+1), & (1) & ; & a=2 b+5, \quad(2) \\
3 \mid(a+b), & (3) & ; & a+7 b \quad \text { is a prime number. }
\end{array}
\]
}

Assume (2) is false. Then on the one hand, according to (4), \(a+7 b \geq 8\) must be prime, while on the other hand, due to (3), \(a+7 b=(a+b)+6 b\) must also be divisible by 3. Therefore, (2) is true.

Then \(a+b=3 b+5\) is not divisible by three, so (3) is false. Therefore, (1) and (4) must be true.

From (2), it follows that \(a+1=2 b+6\), and with (1), this is satisfied if and only if \(b \mid 6\), i.e., \(b \in\{1,2,3,6\}\).

Due to (2), \(a+7 b=9 b+5\). Therefore, \(a+7 b\) can only be prime if \(b\) is even. For \(b=2\), this is the case, since \(9 \cdot 2+5=23\) is prime. For \(b=6\) as well: \(9 \cdot 6+5=59\) is prime.

Thus, \((a, b)=(9,2)\) and \((a, b)=(17,6)\) are all the desired pairs.