2. Provide an example of a set $M$ of two-digit numbers that has the maximum number of elements and at the same time satisfies both of the following conditions:

(i) Any two numbers from M are coprime.

(ii) If we change the order of the digits of any number from M, we get another number from the set M.
2. Due to condition (i), the set M can contain at most one of the numbers $11,22,33, \ldots, 99$ written with two identical digits, which are all divisible by eleven. Due to condition (ii) and divisibility by two, there cannot be any number in M written with two different even digits; with one even digit, M can contain at most one pair of numbers $\overline{a b}, \overline{b a}$.

What remains is to assess how many pairs of numbers $\overline{a b}, \overline{b a}$ written with two different odd digits $a$ and $b$ the set M can contain. None of these numbers can be divisible by three (if the number $\overline{a b}$ is divisible by three, then the number $\overline{b a}$ is also divisible by three), so only seven pairs of such numbers are possible: $(13,31),(17,71),(19,91),(35,53),(37,73),(59,95)$, and $(79,97)$. Due to divisibility by five, seven, and nineteen, however, M can contain only one of the pairs $(19,91),(35,53)$, and $(59,95)$, so at most five of the seven listed pairs.

In total, we find that the set $M$ can contain at most $1+2+2 \cdot 5=13$ numbers. An example of a thirteen-element set is

$$
M=\{11,23,32,13,31,17,71,35,53,37,73,79,97\}
$$

(There are other examples, but our considerations show that any thirteen-element set $M$ must contain the numbers $13,31,17,71,37,73,79,97$ and one of the pairs $(35,53)$ or $(59,95)$; the pair $(19,91)$ is excluded because the number 91 is a multiple of 13.)

For a complete solution, award 6 points, of which 3 points for an example of a thirteen-element set $M$ and 3 points for the justification of why the set $M$ cannot contain more than 13 numbers (in our approach, 1 point for considerations about numbers divisible by eleven and numbers with even digits, 2 points for the discussion of numbers with odd digits).