$A, B, C, D$ four people attended a meeting, and they all received the same positive integer. Next, each person described this positive integer, and each person said three sentences, of which at least one is true and at least one is false. What they said is as follows:
$\mathrm{A}:\left(A_{1}\right)$ This number is less than 12
$\left(A_{2}\right) 7$ cannot divide this number
$\left(A_{3}\right) 5$ multiplied by this number is less than 70
$B:\left(B_{1}\right) 12$ multiplied by this number is greater than 1000
$\left(B_{2}\right) 10$ can divide this number
$\left(B_{3}\right)$ This number is greater than 100
$C:\left(C_{1}\right) 4$ can divide this number
$\left(C_{2}\right) 11$ multiplied by this number is less than 1000
$\left(C_{3}\right) 9$ can divide this number
$D:\left(D_{1}\right)$ This number is less than 20
$\left(D_{2}\right)$ This number is a prime number
$\left(D_{3}\right) 7$ can divide this number
This number is
Let this number be $n$.
We organize the conditions as follows:
$A:\left(A_{1}\right) n1000$, i.e., $n>83 \frac{1}{3} ;\left(B_{2}\right) 10 \mid n ;\left(B_{3}\right) n>100$
$C:\left(C_{1}\right) 4\left|n ;\left(C_{2}\right) 11 n<1000, n<90 \frac{10}{11} ;\left(C_{3}\right) 9\right| n$
$D:\left(D_{1}\right) n<20 ;\left(D_{2}\right) n$ is a prime number; $\left(D_{3}\right) 7 \mid n$
First, we notice that $\left(A_{2}\right)$ and $\left(D_{3}\right)$ are completely opposite.
(1) If $\left(D_{3}\right)$ is true, $\left(A_{2}\right)$ is false; i.e., $7 \mid n$
$\left(A_{1}\right)$ and $\left(A_{3}\right)$ must have at least one true statement.
But it cannot be that $\left(A_{1}\right)$ is true and $\left(A_{3}\right)$ is false.
Therefore, $\left(A_{3}\right)$ is true; then $n<14$

Thus, $\left(D_{1}\right)$ is true.

Therefore, $\left(D_{2}\right)$ is false.
Notice that the only positive integer that satisfies $n<14$ and $7 \mid n$ is $n=7$, and 7 is a prime number, which contradicts the statement “$\left(D_{2}\right) n$ is a prime number” being false.
(2) Therefore, if $\left(D_{3}\right)$ is false, $\left(A_{2}\right)$ is true; i.e., 7 does not divide $n$.
$\left(A_{1}\right)$ and $\left(A_{3}\right)$ must have at least one false statement.

But it cannot be that $\left(A_{1}\right)$ is true and $\left(A_{3}\right)$ is false.

Therefore, $\left(A_{1}\right)$ is false; then $n \geq 12$.
Again, if $\left(A_{1}\right)$ is false, then $n \geq 12$.
Now, let's discuss $\left(D_{1}\right)$.
(1) If $\left(D_{1}\right)$ is true, i.e., $12 \leq n<20$

Therefore, $\left(B_{1}\right)$ and $\left(B_{3}\right)$ are both false.
Then $\left(B_{2}\right)$ is true; then $10 \mid n$
But there is no positive integer $n$ that satisfies both $12 \leq n<20$ and $10 \mid n$.
(2) Therefore, $\left(D_{1}\right)$ is false, i.e., $n \geq 20$.

Then $\left(D_{2}\right)$ is true, i.e., $n$ is a prime number.
Therefore, $\left(B_{2}\right)$, $\left(C_{1}\right)$, and $\left(C_{3}\right)$ are all false.
Then $\left(C_{2}\right)$ is true, i.e., $20 \leq n \leq 90$.
Therefore, $\left(B_{3}\right)$ is false.

Then $\left(B_{1}\right)$ is true, i.e., $84 \leq n \leq 90$.
Notice that the only number that satisfies $84 \leq n \leq 90$ and is a prime number is $n=89$, i.e., this number is 89.