## Task 1 - 260621

In the following five-digit number, two digits have become illegible and have been replaced by asterisks.

$$
27 * * 7
$$

Instead of the asterisks, two digits are to be inserted so that the number is divisible by 9. Give all five-digit numbers that can arise from such insertion! Verify that all the required numbers have been provided by you!
By inserting two digits in place of the asterisks, a number divisible by 9 is created if and only if its cross sum is divisible by 9. The latter is the case if and only if the sum of the two inserted digits is 2 or 11.

Consequently, a number divisible by 9 is created if and only if the two asterisks are replaced in this order by the digits 0,2 or 2,9 or 1,1 or 3,8 or 2,0 or 4,7 or 5,6 or 7,4 or 8,3 or 9,2.

Thus, the sought-after numbers are

27027, 27117, 27207, 27297, 27387, 27477, 27567, 27657, 27747, 27837, and 27927.