10. (10 points) The teacher used the ten digits from 0 to 9 to form five two-digit numbers, with each digit used exactly once; then gave these five two-digit numbers to five clever and honest students $A, B, C, D, E$, each student could only see their own two-digit number, and the following conversation took place:
$A$ said: “My number is the smallest, and it is a prime number.”
$B$ said: “My number is a perfect square.”
$C$ said: “My number is the second smallest, and it has exactly 6 factors.”
$D$ said: “My number is not the largest, and I already know two of the numbers that $A, B, C$ have.”
$E$ said: “My number is three times someone else’s number.”
What is the sum of these five two-digit numbers?
180
【Solution】Solution: $A$ can determine that their number is the smallest, which means the tens digit of $A$ is 1. Since it is also a prime number, $A$ could be $13, 17, 19$;
$C$ can determine that their number is the second smallest, and it has 6 factors, so it could be $20, 28, 32$;
$B$ is a perfect square but cannot contain $1$ or $2$, so $B$ could be $36, 49, 64$;
$D$ can just determine two of the numbers among $A, B, C$, and through experimentation, when $D=36, 37, 39, 40, 47, 48, 49$, it can deduce two of the numbers among $A, B, C$, as follows:
\begin{tabular}{cccc}
$D$ & $A$ & $B$ & $C$ \\
36 & 17 & 49 & 20,28 \\
37 & 19 & 64 & 20,28 \\
39 & 17 & 64 & 20,28 \\
40 & 17,19 & 36 & 28 \\
47 & 19 & 36 & 20,28 \\
48 & 17,19 & 36 & 20 \\
49 & 17 & 36 & 20,28
\end{tabular}

It is found that in any case, 5 does not appear, so $E$ must have a digit 5;
$E$ says their number is three times someone else's number, and the only number that can form a multiple of 3 with 5 and has not been used is 7, so $E$'s number can only be 57 or 75. Clearly, $75 \div 3=25$ does not appear in the table above, so $E$'s number is 57. Therefore, the numbers held by $A, B, C, D, E$ have the following two possibilities: $19, 36, 28, 40, 57$ or $19, 36, 20, 48, 57$,
$$
\begin{array}{l}
19+36+28+40+57=180 \\
19+36+20+48+57=180
\end{array}
$$

Answer: The sum of these five two-digit numbers is 180.
Therefore, the answer is: 180.