## Task 1 - 330731

Determine all four-digit natural numbers $z$ that satisfy the following conditions (1) and (2)!

(1) The number $z$ is divisible by 48.

(2) The second digit of the number $z$ is a 3, and the third digit of $z$ is a 4.
I. If a four-digit natural number $z$ satisfies conditions (1) and (2), then:

According to (1) and since $48=2 \cdot 8 \cdot 3$, $z$ is also divisible by 8 and by 3. A number is only divisible by 8 if the number represented by its last three digits is divisible by 8. Among the numbers from 340 to 349, only the number 344 is divisible by 8. Therefore, and due to (2), the last three digits of $z$ can only be $3,4,4$ in this order.

A number is only divisible by 3 if its cross sum is divisible by 3. Since the sum of the last three digits $3+4+4=11$, a cross sum divisible by 3 can only be achieved with the first digit if it is 1, 4, or 7.

The number 4344 is (since the division 4344 : 48 results in 90 with a remainder of 24) not divisible by 48, so the 4 as the first digit is excluded.

Thus, among the four-digit natural numbers, only the two numbers 1344 and 7344 can satisfy conditions (1) and (2).

II. These two numbers obviously satisfy (2) and (since $1344: 48=28$ and $7344: 48=153$) also (1).

Thus, all numbers of the required type have been determined: They are exactly the two numbers 1344 and 7344.