10. (5 points) There are 10 distinct natural numbers, the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is the minimum sum of these 10 natural numbers?
【Analysis】According to the problem, the product of any 5 numbers is even, which means there are at most 4 odd numbers. Since the sum of all 10 numbers is odd, the number of odd numbers must be 1 or 3. Taking the smallest odd numbers 1 or 1, 3, 5, and comparing the possible cases of the other numbers, we can find the minimum value.

【Solution】According to the analysis, the number of odd numbers among the 10 natural numbers is 1 or 3,
(1) If there is only one odd number, then the smallest odd number is 1, and the smallest even numbers are: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18,
At this time, the sum of the natural numbers = 0 + 1 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 91;
(2) If there are three odd numbers, then the odd numbers are 1, 3, 5, and the smallest even numbers are: 0, 2, 4, 6, 8, 10, 12,
At this time, the sum of the natural numbers = 0 + 1 + 3 + 5 + 2 + 4 + 6 + 8 + 10 + 12 = 51.

In summary, the minimum sum of these 10 natural numbers is 51.
Therefore, the answer is: 51.