6.5. Ten different natural numbers are such that the product of any five of them is even, and the sum of all ten is odd. What is the smallest possible sum of all these numbers? Justify your answer.
Solution: There cannot be more than four odd numbers (five odd numbers in a product give an odd number), and, moreover, their quantity must be odd. Therefore, there are either 3 or 1 odd numbers, and 7 or 9 even numbers, respectively. The sets with the smallest sum are 1, 3, 5, 2, 4, 6, 8, 10, 12, 14 and 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, respectively. The latter set gives a larger sum, so the first one fits. The sum of the numbers in it is 51.

Answer: 51.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| Correct and justified set found, but the sum of its numbers is calculated incorrectly | 6 points |
| Proven that the number of odd numbers in the set is odd, but not all such cases are correctly investigated. | 3 points |
| The answer is correct, a set realizing this answer is provided, but the optimality of the provided example is not proven | 2 points |
| The correct set is indicated, but its optimality is not proven, and the sum of the numbers in the set is calculated with an error | 1 point |
| Answer without justification and/or incorrect answer | 0 points |