## Task 2 - 180622

Determine all two-digit numbers $z$ that simultaneously satisfy the following conditions (1), (2):

(1) The units digit of $z$ is 1 less than the tens digit of $z$.

(2) If the digits of $z$ are swapped, the result is a two-digit prime number.
If $z$ is a number with the required properties, then $z$ does not have 0 as its unit digit according to (2), so the unit digit is one of the digits $1, \ldots, 9$.

Since the tens digit is 1 greater according to (1), 9 is excluded as the unit digit, and due to (1), for the two-digit numbers $z$, only the possibilities 21, 32, 43, 54, 65, 76, 87, 98 remain.

Of these, 21, 43, 65, and 87 are excluded because swapping the digits results in an even two-digit number, which is not a prime number. Furthermore, the number 54 is excluded because swapping the digits results in the two-digit number 45, which is divisible by 5.

Therefore, only the numbers 32, 76, and 98 can satisfy all the conditions of the problem. Indeed, they are two-digit numbers and satisfy (1), and they also satisfy (2) because 23, 67, and 89 are two-digit prime numbers. The numbers sought are thus 32, 76, and 98.