9.5. What is the largest number of different natural numbers that can be chosen so that the sum of any three of them is a prime number?
Answer: 4 numbers. Example: $1,3,7,9$. Indeed, the numbers $1+3+7=11, 1+3+9=13$, $1+7+9=17, 3+7+9=19$ are prime. Evaluation. Note that among five natural numbers, it is always possible to choose three whose sum is a composite number. Consider the remainders of these five numbers when divided by 3. If there are three identical remainders, then the sum of the corresponding numbers is divisible by 3. If there are no three identical remainders, then each of the three possible remainders $0,1,2$ must be present. Then the sum of three numbers with different remainders when divided by 3 is divisible by 3. This sum is not equal to 3, since all numbers are different and natural. Therefore, this sum is a composite number.

Remarks. An example of four numbers without evaluation - no more than 3 points.