## Task 3 - 100623

In the five-digit number $52 * 2 *$, two (same or different) digits are to be inserted at the positions marked with * so that the resulting number is divisible by 36.

List all possibilities for this!

(Note: A number is divisible by 36 if and only if it is divisible by 4 and by 9.)
(1) If after supplementing the two missing digits a number divisible by 36 has been formed, then this number is divisible by both 4 and 9.

(2) If a multi-digit number is divisible by 4, then its last two digits (in the same order) also form a number divisible by 4. Therefore, the units digit can only be 0, 4, or 8.

(3) If a number is divisible by 9, then so is its cross sum. Now, the sum of the three given digits is 9. If the units digit is $\left\{\begin{array}{l}0 \\ 4 \\ 8\end{array}\right\}$, then the hundreds digit can only be $\left\{\begin{array}{l}0 \text { or } 9 \\ 5 \\ 1\end{array}\right\}$.

Therefore, the numbers 52020; 52920; 52524; 52128 can only satisfy the conditions of the problem. The check shows that they all do.