9.2. We will call a natural number interesting if the sum of its digits is a prime number. What is the maximum number of interesting numbers that can be among five consecutive natural numbers?

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# Answer: 4.

Solution. Among five consecutive natural numbers, there can be 4 interesting numbers. For example, the numbers 199, 200, 201, 202, 203 (with digit sums 19, 2, 3, 4, and 5) will work.

Now, let's prove that all 5 numbers cannot be interesting. Among our five numbers, there are three that lie within the same decade. Then their digit sums are consecutive numbers; hence, they cannot all be prime simultaneously.

Remark 1. There are other examples of five consecutive numbers, four of which are interesting. Moreover, the number divisible by 10 can also be in a different position, as in the set 197, 198, 199, 200, 201. However, in any such example, the digit sums of two of the numbers in the set must be 2 and 3.

Remark 2. One can also prove that five consecutive numbers cannot all be interesting in a different way. Among our numbers, at least two have even digit sums (even if there is a transition "across a decade"). But both of these sums cannot equal 2, since the digit sum 2 is only found in numbers of the form $10 \ldots 010 \ldots 0$ and $20 \ldots 0$, and two such numbers cannot appear among 5 consecutive natural numbers.

Comment. Only the answer - 0 points.

Providing an example of five consecutive numbers, four of which are interesting - 3 points.

Providing only an example of five consecutive numbers, among which fewer than 4 are interesting - 0 points.

Proving that there do not exist five consecutive interesting numbers - 3 points.