7. Append a two-digit ending to the number 2017 so that the resulting six-digit number is divisible by 72.
First method:

A natural number is divisible by 72 if it is divisible by both 8 and 9.

A natural number is divisible by 8 if its last three digits are divisible by 8.

A natural number is divisible by 9 if the sum of all its digits is divisible by 9.

The first natural number greater than 201700 that is divisible by 8 is 201704 because $201700 \div 8 = 25212$ with a remainder of 4.

1 POINT

All suitable two-digit endings are given in the table:

2 POINTS

| two-digit ending (divisibility by 8) | number | sum of digits (divisibility by 9) |  |
| :---: | :---: | :---: | :---: |
| 04 | 201704 | 14 |  |
| 12 | 201712 | 13 |  |
| 20 | 201720 | 12 |  |
| 28 | 201728 | 20 |  |
| 36 | 201736 | 19 |  |
| 44 | 201744 | 18 | 2 POINTS |
| 52 | 201752 | 17 |  |
| 60 | 201760 | 16 |  |
| 68 | 201768 | 24 |  |
| 76 | 201776 | 23 |  |
| 84 | 201784 | 22 |  |
| 92 | 201792 | 21 |  |

The desired six-digit number is 201744.

1 POINT

TOTAL 10 POINTS