1. Find the largest three-digit number from which, after erasing any digit, we get a prime number.
1. We are looking for the largest number $\overline{a b c}$, for which all three numbers $\overline{a b}, \overline{a c}$, and $\overline{b c}$ are prime numbers. The two-digit prime numbers $\overline{a b}$ and $\overline{a c}$ are odd, so their last digits must be odd. Additionally, none of them can contain the digit 5, otherwise the mentioned two-digit numbers would be divisible by five. Therefore, it is sufficient to further examine only the digits $b$ and $c$ from the set $\{1,3,7,9\}$.

We want the digit $a$ to be as large as possible, so we will sequentially check the possibilities starting with the value $a=9$, until we find a solution.

For $a=9$, the numbers $91=7 \cdot 13, 93=3 \cdot 31, 99=3 \cdot 33$ are composite, and only 97 is a prime number. Thus, the only possibility is $b=c=7$. However, in this case, the number $\overline{b c}=77=7 \cdot 11$ is composite. Therefore, the desired number cannot start with the digit 9.

Similarly, for $a=8$, we rule out the possibilities $b, c \in\{1,7\}$, because $81=3 \cdot 27$ and $87=3 \cdot 29$. This leaves only the possibility $b, c \in\{3,9\}$. In that case, however, the number $\overline{b c}$ is divisible by three.

If the desired number starts with the digit $a=7$, $b$ cannot be 7 nor can $c$ be 7, but the numbers 71, 73, and 79 are all prime numbers. From the remaining candidates $1,3,9$ for the digits $b, c$, four two-digit prime numbers can be formed: $31, 19, 13, 11$, and the largest of them is 31. We see, therefore, that the number 731 satisfies the conditions of the problem, and no larger such three-digit number exists.

The largest three-digit number sought is 731.

Award 6 points for a complete solution. Award 1 point for ruling out the possibilities $b, c \in\{2,4,6,8\}$, and another point for ruling out $b=5$ and $c=5$. Award 1 point for each of the possibilities $a=9$ and $a=8$ that are ruled out. Award the last 2 points for examining the possibility $a=7$ and finding the correct solution. Systematic searching of possibilities from the largest three-digit number without finding the correct solution should be rated no higher than 4 points.