$4 \cdot 11$ The math teacher told $S$ the number of divisors of a two-digit natural number $n$, and told $P$ the sum of the digits of $n$. $S$ and $P$ are two very smart students who hope to deduce the exact value of $n$. $S$ and $P$ had the following conversation.
$P:$ I don't know what $n$ is.
$S$ : I don't know either, but I know whether $n$ is even.
$P$ : Now I know what $n$ is.
$S$ : Now I also know.
The teacher confirmed that $S$ and $P$ are honest and trustworthy, and every word they said is based on evidence.
What is the value of $n$? Why?
(China National Training Team Test Question, 1988)
［Solution］Let the sum of the digits of $n$ be $p$, and the number of divisors be $s$.
The first sentence indicates: $2 \leqslant p \leqslant 17$.
This is because if $n=10$, then $p=1$, and if $n=99$, then $p=18$. Therefore, if $P$ knew $p=18$ or $p=1$, they would know what $n$ is.
The second sentence indicates: $s=2,3,8,10,12$.
This is because $s \neq 1$ (otherwise $n=1$), $s \neq 11$ (otherwise $n=2^{10}$ is not a two-digit number).

Also, $s \leqslant 12$, otherwise if $s=13$, $2^{12}$ is not a two-digit number. If $s>13$, $n$ cannot be a two-digit number.
If $s=7$, then $n=2^{6}=64$, which contradicts $S$ saying they don't know.
If $s=9$, then $n=2^{2} \cdot 3^{2}=36$, which contradicts $S$ saying they don't know.
If $s=4$, then $n=2 \cdot 7=14$ or $n=3 \cdot 5$, which can be either odd or even.
If $s=5$, then $n=2^{4}=16$ or $n=3^{4}=81$, which can be either odd or even.
If $s=6$, then $n=2^{2} \cdot 3=12$, or $n=3^{2} \cdot 2=18$, or $n=3^{2} \cdot 5=45$.
All of these make it impossible for $S$ to determine whether $n$ is even.
When $s=2$, $n$ must be an odd prime.
When $s=3$, $n$ must be the square of an odd prime.
When $s=8,10,12$, $n$ must be even.
The third sentence indicates: $n=11,30,59,89$, with the corresponding $p=2,3,14,17$.
This is because for $p=4,5,6,7,8,9,10,11,12,13,15,16$, there are two values each that fit $s=2,3,8,10$ or 12.
For example, 13 or 31, 23 or 41, 24 or 42, 43 or 61, 17 or 71, 54 or 72, 37 or 73, 29 or 83; 48 or 84, 49 or 67, 78 or 96, 79 or 97.
For $p=2,3,14,17$, there is only one value of $n$ that fits $s=2,3,8,10$, or 12, i.e., $n=11,30,59$ or 89 (in fact, at this point, only $s=2$ and 8).
The fourth sentence indicates: $n=30$.
This is because from the third sentence, when $s=2$, $n$ can be 11, 59, and 89, and $S$ cannot determine the value of $n$, so only $s=8$, i.e., $n=30$.