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If \(x\) and \(y\) are integers and \((x - 1)(y + 3)(xy + 3) = 0\), which of the following must be true?
I. If \(x = -1\), \(|y|\) is a prime number
II. If \(|y|\) is not a prime number, \(x\) is not a prime number
III. If \(x\) is a prime number, \(\frac{y}{x}\) is not an integer
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
I. If \(x = -1\), \(|y|\) is a prime number
II. If \(|y|\) is not a prime number, \(x\) is not a prime number
III. If \(x\) is a prime number, \(\frac{y}{x}\) is not an integer
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
07 Aug 2026, 05:21
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Official Solution:
If \(x\) and \(y\) are integers and \((x - 1)(y + 3)(xy + 3) = 0\), which of the following must be true?
I. If \(x = -1\), \(|y|\) is a prime number
II. If \(|y|\) is not a prime number, \(x\) is not a prime number
III. If \(x\) is a prime number, \(\frac{y}{x}\) is not an integer
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Since \((x - 1)(y + 3)(xy + 3) = 0\), at least one of the following must be true:
• \(x = 1\)
• \(y = -3\)
• \(xy = -3\). Since \(x\) and \(y\) are integers, \(xy = -3\) gives:
\(x = 3\) and \(y = -1\)
\(x = -3\) and \(y = 1\)
\(x = 1\) and \(y = -3\)
\(x = -1\) and \(y = 3\)
So the equation can be satisfied if at least one of the following is true:
1. \(x = 1\)
2. \(y = -3\)
3. \(x = 3\) and \(y = -1\)
4. \(x = -3\) and \(y = 1\)
5. \(x = 1\) and \(y = -3\)
6. \(x = -1\) and \(y = 3\)
Now check the statements.
I. If \(x = -1\), \(|y|\) is a prime number
If \(x = -1\), cases 1, 3, 4, and 5 are impossible. So either \(y = -3\) from case 2, or \(x = -1\) and \(y = 3\) from case 6. In either case, \(|y| = 3\), which is prime.
Therefore, statement I must be true.
II. If \(|y|\) is not a prime number, \(x\) is not a prime number
If \(|y|\) is not a prime number, cases 2, 5, and 6 are impossible. However, case 3, \(x = 3\) and \(y = -1\), is still possible. Here, \(|y| = 1\), which is not prime, but \(x = 3\), which is prime.
Therefore, statement II is not necessarily true.
III. If \(x\) is a prime number, \(\frac{y}{x}\) is not an integer
If \(x\) is a prime number, cases 1, 4, and 6 are impossible, since \(x\) is not prime in those cases. However, case 2, \(y = -3\), is still possible. And if \(y = -3\), \(x\) can be any prime. For example, if \(x = 3\) and \(y = -3\), then \(\frac{y}{x} = \frac{-3}{3} = -1\), which is an integer.
Therefore, statement III is not necessarily true.
Answer: A
If \(x\) and \(y\) are integers and \((x - 1)(y + 3)(xy + 3) = 0\), which of the following must be true?
I. If \(x = -1\), \(|y|\) is a prime number
II. If \(|y|\) is not a prime number, \(x\) is not a prime number
III. If \(x\) is a prime number, \(\frac{y}{x}\) is not an integer
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Since \((x - 1)(y + 3)(xy + 3) = 0\), at least one of the following must be true:
• \(x = 1\)
• \(y = -3\)
• \(xy = -3\). Since \(x\) and \(y\) are integers, \(xy = -3\) gives:
\(x = 3\) and \(y = -1\)
\(x = -3\) and \(y = 1\)
\(x = 1\) and \(y = -3\)
\(x = -1\) and \(y = 3\)
So the equation can be satisfied if at least one of the following is true:
1. \(x = 1\)
2. \(y = -3\)
3. \(x = 3\) and \(y = -1\)
4. \(x = -3\) and \(y = 1\)
5. \(x = 1\) and \(y = -3\)
6. \(x = -1\) and \(y = 3\)
Now check the statements.
I. If \(x = -1\), \(|y|\) is a prime number
If \(x = -1\), cases 1, 3, 4, and 5 are impossible. So either \(y = -3\) from case 2, or \(x = -1\) and \(y = 3\) from case 6. In either case, \(|y| = 3\), which is prime.
Therefore, statement I must be true.
II. If \(|y|\) is not a prime number, \(x\) is not a prime number
If \(|y|\) is not a prime number, cases 2, 5, and 6 are impossible. However, case 3, \(x = 3\) and \(y = -1\), is still possible. Here, \(|y| = 1\), which is not prime, but \(x = 3\), which is prime.
Therefore, statement II is not necessarily true.
III. If \(x\) is a prime number, \(\frac{y}{x}\) is not an integer
If \(x\) is a prime number, cases 1, 4, and 6 are impossible, since \(x\) is not prime in those cases. However, case 2, \(y = -3\), is still possible. And if \(y = -3\), \(x\) can be any prime. For example, if \(x = 3\) and \(y = -3\), then \(\frac{y}{x} = \frac{-3}{3} = -1\), which is an integer.
Therefore, statement III is not necessarily true.
Answer: A
