## Task 36/72

We are looking for all four-digit prime numbers with the following properties:

1. All digits in the decimal representation are different from each other.
2. If the number is split in the middle into two two-digit numbers, both numbers are prime numbers, each with a digit sum of 10.
3. The last two digits are also prime numbers, each on their own.
We first determine the two-digit numbers whose cross sum is 10:

$$
19 ; 28 ; 37 ; 46 ; 55 ; 64 ; 73 ; 82 ; 91
$$

Among them, only three are prime numbers: 19; $37 ; 73$.

Since 1 and 9 are not prime numbers, the second of the two-digit prime numbers can only be 37 and 73. Then, due to condition 1, the first of the two-digit prime numbers must be the number 19. Thus, the two possible numbers are 1937 and 1973; however, 1937 is not a prime number: $1937=13 \cdot 449$. Therefore, the only solution left is the prime number 1973.

### 2.13 Problems and Solutions 1973