## Task 2 - 180612

A number $z$ is to be written in the form $z=\star 3 \star 60$, where each star $(\star)$ is to be replaced by one of the digits 0 to 9 so that $z$ has the following two properties:

(1) $60000<z<100000$,

(2) $z$ is divisible by 9.

Determine all numbers $z$ that satisfy these conditions!
If $z$ satisfies the conditions, then due to (1), the ten-thousands digit of $z$ is one of the numbers $6, 7, 8, 9$.

Furthermore, $z$ is divisible by 9 because of (2), and therefore the sum of the digits of $z$ is also divisible by 9. Since the sum of the thousands, tens, and units digits is 9, the hundreds digit must complement the above-mentioned ten-thousands digit $6, 7, 8$, or 9 to form a number divisible by 9.

This is only the case for the numbers 63360, 73260, 83160, 93060, 93960.

Each of these numbers satisfies (1) and, since they are divisible by 9, also (2). Therefore, the numbers given are all the ones sought.