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20 May 2024, 01:30
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B
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Difficulty:
35%
(medium)
Question Stats:
68% (01:30) correct
32%
(01:42)
wrong
based on 196
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If \(x\) is a positive integer and the product of any two distinct positive factors of \(x\) is odd, which of the following must be true?
I. \(x\) is an odd prime.
II. \(x^2\) is odd.
III. \(x\) is the square of an odd number.
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
I. \(x\) is an odd prime.
II. \(x^2\) is odd.
III. \(x\) is the square of an odd number.
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
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02 Jun 2024, 14:15
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Official Solution:
If \(x\) is a positive integer and the product of any two distinct positive factors of \(x\) is odd, which of the following must be true?
I. \(x\) is an odd prime.
II. \(x^2\) is odd.
III. \(x\) is the square of an odd number.
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Given that the product of any two distinct positive factors of \(x\) is odd, \(x\) cannot be even because if it is, then it has 1 and 2 as factors and \(1 * 2 = 2 = even\). Thus, \(x\) must be odd. In this case, all its factors will be odd and the product of any two distinct positive factors of \(x\) will be odd. Therefore, only II must be true because if \(x\) is odd, then \(x^2\) will also be odd.
Answer: B
If \(x\) is a positive integer and the product of any two distinct positive factors of \(x\) is odd, which of the following must be true?
I. \(x\) is an odd prime.
II. \(x^2\) is odd.
III. \(x\) is the square of an odd number.
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Given that the product of any two distinct positive factors of \(x\) is odd, \(x\) cannot be even because if it is, then it has 1 and 2 as factors and \(1 * 2 = 2 = even\). Thus, \(x\) must be odd. In this case, all its factors will be odd and the product of any two distinct positive factors of \(x\) will be odd. Therefore, only II must be true because if \(x\) is odd, then \(x^2\) will also be odd.
Answer: B
General Discussion
20 May 2024, 01:42
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Bunuel
Quote:
Inference: x is an odd number, as all the factors of x are odd.
Answer choice elimination
I. \(x\) is an odd prime.
All we know is that x is an odd integer. x could be a prime number, however, it is not necessary that \(x\) is a prime number.
II. \(x^2\) is odd.
We know that \(x\) is an odd integer. Hence, \(x^2\) is also odd.
III. \(x\) is the square of an odd number.
Just like I, this could be true. However, the statement is not always true.
For example: if x = 3, x is not a square of an odd number.
Option B
