6.5. On cards, all two-digit numbers from 10 to 99 are written (one number per card). All these cards are lying on the table face down. What is the minimum number of cards that need to be flipped to guarantee that at least one of the revealed numbers is divisible by 7? Justify your answer.
Solution: There are exactly 13 two-digit numbers divisible by 7 (14 = 7 * 2, 21 = 7 * 3, ..., 98 = 7 * 14). If at least 13 cards are not flipped, it is impossible to exclude the situation where all these numbers remain covered. Therefore, no more than 12 cards can be left in their original position, and the rest must be flipped. Since there are 90 cards in total, at least 90 - 12 = 78 cards should be flipped.

Answer: 78 cards.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer | 7 points |
| Correctly and justifiedly found the maximum number of cards that can be left unflipped | 5 points |
| Correct reasoning but incorrect answer due to incorrect total number of two-digit numbers found | 4 points |
| Incorrect number of two-digit numbers divisible by 7 found, which may have led to an incorrect answer | 3 points |
| There is an idea (not developed into a solution) to find the number of cards with a number divisible by 7 | 2 points |
| Correct answer without justification | 1 point |
| Fundamentally incorrect reasoning | 0 points |