## Task 2 - 330722

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

(1) The number $z$ is divisible by 24.

(2) The second digit of the number $z$ is a 1, and the third digit of $z$ is a 3.
Condition (1) is satisfied if and only if the number $z$ is divisible by 3 and by 8; since 3 and 8 are coprime, and $3 \cdot 8 = 24$.

The number $z$ is divisible by 8 if and only if the number represented by its last three digits is divisible by 8. Together with condition (2), this is the case if and only if the fourth digit of $z$ is 6; because among the numbers from 130 to 139, only the number 136 is divisible by 8.

The number $z$ is divisible by 3 if and only if its cross sum is divisible by 3.

Together with the already mentioned conditions, this is the case if and only if the first digit of $z$ is one of the three digits 2, 5, 8; because the sum of the last three digits 1, 3, 6 is 10, and among the sums from 11 to 19 (resulting from adding another digit 1,..., 9) exactly 12, 15, and 18 are divisible by 3.

Thus, exactly the numbers 2136, 5136, 8136 satisfy conditions (1) and (2).