## Task 1 - 320621

In the following six-digit number, two digits have become illegible and have been replaced by asterisks:

$$
38 \star \star 42
$$

Instead of the asterisks, two digits are to be inserted so that the number is divisible by 9. Give all six-digit numbers that can arise from such insertion! Verify that all the required numbers have been provided by you!
By inserting two digits, a number divisible by 9 is created if and only if its cross sum is divisible by 9. This is the case when $3+8+4+2=17$ if the sum of the two inserted digits differs from 1 by a multiple of the number 9.

Since the sum of two digits can be at most $9+9=18$, it follows that the sum of the digits to be inserted can only be 1 or 10. All possibilities for forming one of these sums from two digits are

$$
\begin{aligned}
& 1=0+1=1+0 \\
& 10=1+9=2+8=3+7=4+6=5+5=6+4=7+3=8+2=9+1
\end{aligned}
$$

Therefore, all the desired numbers are

$380142,381042,381942,382842,383742,384642,385542,386442,387342,388242,389142$