\section*{Task 1 - 131021}

Determine all (in the decimal number system) three-digit prime numbers with the following properties!

(1) If each digit of the three-digit prime number is written separately, each one represents a prime number.

(2) The first two and the last two digits of the three-digit prime number represent (in this order) a two-digit prime number each.
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Since a multi-digit prime number can only end in 3 or 7, due to (1) the last digit and due to (2) also the middle digit must lie in \(\{3,7\}\). The number formed by the last two digits must therefore be 37 or 73, as the other two possibilities 33 and 77 are divisible by 11, which contradicts condition (2).

Thus, for the first digit, which according to (1) must lie in \(\{2,3,5,7\}\), only 3 or 7 remain as options, since otherwise the sum of the digits and thus the number itself would be divisible by 3.

From what has been proven so far, all digits of the number must therefore lie in \(\{3,7\}\), and no two consecutive digits can be the same, as this would contradict condition (2). Of the two remaining possibilities, 373 and 737, the latter is divisible by 11 and thus not prime, while 373 is the only number that satisfies all the conditions here.