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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, 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, 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
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M46-47
16 Jul 2026, 09:00
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GMAT Club 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, 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:
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, case 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, 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 y/x = -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, 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
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, case 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, 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 y/x = -3/3 = -1, which is an integer.
Therefore, statement III is not necessarily true.
Answer: A.
General Discussion
16 Jul 2026, 08:34
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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?
must be true condition
I. If x = -1, |y| is a prime number ; this condition yes y has to be -3 and lyl will be 3
must be true ; correct option
II. If |y| is not a prime number, x is not a prime number
x has to be 1 to make equation= 0 , so lyl can be either prime or not a prime , both cases possible y can be -3 or 4 ; in correct option
III. If x is a prime number, y/x is not an integer
if x = 3 and y = -3 ; this is not true , incorrect option
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
OPTION A ; is correct for must be true condition
must be true condition
I. If x = -1, |y| is a prime number ; this condition yes y has to be -3 and lyl will be 3
must be true ; correct option
II. If |y| is not a prime number, x is not a prime number
x has to be 1 to make equation= 0 , so lyl can be either prime or not a prime , both cases possible y can be -3 or 4 ; in correct option
III. If x is a prime number, y/x is not an integer
if x = 3 and y = -3 ; this is not true , incorrect option
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
OPTION A ; is correct for must be true condition
