9.2. On the board, a certain natural number $N$ was written nine times (one under the other). Petya appended a non-zero digit to the left or right of each of the 9 numbers; all the appended digits are different. What is the maximum number of prime numbers that could result among the 9 obtained numbers?

(I. Efremov)
Answer: 6.

Solution. Let $S$ be the sum of the digits of the number $N$. Then the sums of the digits of the obtained numbers will be $S+1, S+2, \ldots, S+9$. Three of these sums will be divisible by 3. By the divisibility rule for 3, the corresponding three numbers on the board will also be divisible by 3. Since these numbers will be greater than 3, they will be composite. Therefore, more than 6 prime numbers cannot appear on the board.

Six prime numbers can appear even when $N=1$ - for example, if Petya gets, among others, the numbers 11, 13, 41, 61, 17, and 19.

Remark. Petya can get six prime numbers even if he only appends digits to one side, for example, from the number $N=3$ he can get the numbers 13, 23, 43, 53, 73, and 83.

Comment. Only an example is provided where 6 prime numbers appeared on the board - 2 points.

Only proven that more than six prime numbers could not appear - 5 points.