## Task 3 - 190723

Determine all natural numbers $z$ that satisfy the following conditions (1) to (4)!

(1) $z$ is a three-digit number.

(2) The tens digit (i.e., the digit in the tens place) of $z$ is 1 greater than the hundreds digit of $z$.

(3) The units digit of $z$ is twice as large as the hundreds digit of $z$.

(4) $z$ is double a prime number.
I) If a natural number $z$ satisfies conditions (1) to (4) and $a$ is its hundreds digit, then: Because of (1), $a \neq 0$, and because of (3), $2a < 10$, so $a < 5$. The following table contains the remaining possibilities for $a=1,2,3,4$ and the resulting tens and units digits according to (2) and (3), and thus $z$.

| Hundreds digit $a$ | Tens digit | Units digit | $z$ |
| :---: | :---: | :---: | :---: |
| 1 | 2 | 2 | 122 |
| 2 | 3 | 4 | 234 |
| 3 | 4 | 6 | 346 |
| 4 | 5 | 8 | 458 |

Of these, the number $z=234$ is eliminated because it is double 117, and since $117=3 \cdot 39$, it is not a prime number. Therefore, only the numbers 122, 346, and 458 can satisfy conditions (1) to (4).

II) They are three-digit numbers, thus satisfying (1). Furthermore, the table shows that they satisfy (2) and (3). Finally, they also satisfy (4) because they are each double 61, 173, and 229, respectively, and these numbers are prime.

Thus, the sought numbers are: 122, 346, 458.