\section*{Exercise 3 - 011123}

Meier, Krause, Schulze, and Franke and their wives buy poultry. Each of the 8 people buys as many animals as they pay DM for each animal. Each man spends \(96,-\) DM more than his wife. Meier buys as many animals as his sisters-in-law combined. Krause buys as many as his sister-in-law. The Schulzes together buy twice as many as Krause. Mrs. Schulze is a born Lehmann. What are the maiden names of the other three women?

Note: Here, a brother-in-law (sister-in-law) refers only to the spouses of siblings or the siblings of the spouse.
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Let's denote the number of animals bought with \(a_{1}, a_{2}, a_{3}, a_{4}\) (for Mr. Meier, Krause, Schulze, and Franke) and \(b_{1}, b_{2}, b_{3}, b_{4}\) (for the women in this order), where \(a_{i}, b_{i} \in \mathbb{N}\).

Then each of the men spends \(a_{i}^{2}\) DM and each of the women spends \(b_{i}^{2}\) DM, and it holds that:

\[
a_{i}^{2}-b_{i}^{2}=\left(a_{i}+b_{i}\right)\left(a_{i}-b_{i}\right)=96=2^{5} \cdot 3=\{48 \cdot 2,32 \cdot 3,24 \cdot 4,16 \cdot 6,12 \cdot 8\}
\]

Thus, the following pairs \(\left(a_{i}, b_{i}\right)\) are possible: \((25,23),(14,10),(11,5),(10,2)\).

The statement "Meier buys as many animals as his sisters-in-law together" can only mean that the Meiers are the pair \((25,23)\) and the men of the pairs \((14,10)\) and \((11,5)\) are his sisters-in-law.

The number 10 appears twice, so the pair \((10,2)\) must belong to the Krauses, and the woman of the pair \((14,10)\) is his sister-in-law.

From "Schulzes buy together twice as many as the Krauses," it follows that the pair \((14,10)\) are the Schulzes, and finally, \((11,5)\) are the Frankes.

Thus, Mr. Meier is related by marriage to both Mr. Schulze and Mr. Franke.

This means that in the first case, either Mrs. Meier is a born Schulze or Mrs. Schulze is a born Meier. The latter is excluded because Mrs. Schulze is a born Lehmann, so: Mrs. Meier is a born Schulze.

Mrs. Franke is a born Meier. Mrs. Krause is a born Schulze.