\section*{Problem 1 - 071211}

Four people \(A, B, C, D\) agreed on a positive integer. Each of the four gives three statements about this number, of which at least one is true and at least one is false:

A: 1. The number is divisible by 4;

2. it is divisible by 9;

3. eleven times the number is less than 1000.

B: 1. The number is divisible by 10;

2. it is greater than 100;

3. twelve times the number is greater than 1000.

\(C:\) 1. The number is a prime number;

2. it is divisible by 7;

3. it is less than 20.

D: 1. The number is not divisible by 7;

2. it is less than 12;

3. five times the number is less than 70.

What is the number?
}

To abbreviate, we denote the given statements with \(A_{1}, A_{2}, A_{3}, B_{1}, B_{2}, B_{3}\), etc. We first examine the statements \(C_{1}, C_{2}, C_{3}\). At least one of them is true and at least one is false. This results in the six cases listed in the following table, where for brevity, a true statement is denoted by \(W\) and a false one by \(F\):

\begin{tabular}{|l|llllll|}
\hline & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline\(C_{1}\) & \(W\) & \(W\) & \(W\) & \(F\) & \(F\) & \(F\) \\
\(C_{2}\) & \(W\) & \(F\) & \(F\) & \(W\) & \(W\) & \(F\) \\
\(C_{3}\) & \(F\) & \(W\) & \(F\) & \(W\) & \(F\) & \(W\) \\
\hline
\end{tabular}

1. If \(C_{1}\) and \(C_{2}\) are true, then the thought number \(z\) is a prime number divisible by 7, so \(z=7\). Then \(C_{3}\) is also true, which contradicts the condition. Therefore, this case cannot occur.

2. If \(C_{1}\) and \(C_{3}\) are true and \(C_{2}\) is false, then \(z\) is a prime number not divisible by 7 and \(z>20\). Then the statements \(B_{1}, B_{2}\), and \(B_{3}\) are all false, which contradicts the condition, so this case cannot occur either.

3. If \(C_{1}\) is true and \(C_{2}\) and \(C_{3}\) are false, then \(z\) is a prime number and \(z \geq 20\). Then \(A_{1}\) and \(A_{2}\) are false, so \(A_{3}\) is true, i.e., \(11 z1000, z \geq 84\).

Among the numbers 84, 85, ..., 90, only 89 is a prime number, so \(z=89\). To conclude, one can verify that then \(D_{1}\) is true and \(D_{2}\) and \(D_{3}\) are false, so all conditions are satisfied.

4. If \(C_{2}\) and \(C_{3}\) are true and \(C_{1}\) is false, then \(z=14\). Then \(D_{1}, D_{2}\), and \(D_{3}\) are false, so this case cannot occur.

5. If \(C_{2}\) is true and \(C_{1}\) and \(C_{3}\) are false, then \(z \geq 20\) and \(z\) is divisible by 7. Then \(D_{1}\), \(D_{2}\), and \(D_{3}\) are false, so this case cannot occur either.

6. If \(C_{1}\) and \(C_{2}\) are false and \(C_{3}\) is true, then \(z<20\), \(z\) is not a prime number, and not divisible by 7. Then \(B_{2}\) and \(B_{3}\) are false and \(B_{1}\) is true, so \(z=10\).

Then \(D_{1}, D_{2}\), and \(D_{3}\) are true, so this case cannot occur either.

Therefore, only the 3rd case meets all the given conditions. The number to be determined is 89.

Adapted from \([5]\)