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If \(x\) and \(y\) are integers is \(xy\) a prime number?
(1) \(x^2 = 1\)
(2) \(y\) is a prime number and \(x\) is a positive number but not a prime number.
(1) \(x^2 = 1\)
(2) \(y\) is a prime number and \(x\) is a positive number but not a prime number.
16 Sep 2014, 00:16
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Official Solution:
If \(x\) and \(y\) are integers is \(xy\) a prime number?
Firstly that only positive integers can be prime numbers and that 1 is not a prime number.
Secondly, for the product of two integers \(x\) and \(y\) to be a prime number, either one of them must be \(1\) and the other a prime number, for example (1, 2), (1, 3), (1, 5), (1, 7), ..., or one of them must be \(-1\) and the other \(-\text{prime}\), for example (-1, -2), (-1, -3), (-1, -5), (-1 ,-7), ...
(1) \(x^2=1\).
Either \(x=1\) or \(x=-1\). Not sufficient.
(2) \(y\) is a prime number and \(x\) is a positive number but not a prime number.
If \(y=\text{prime}\) and \(x=1\), then \(xy=y=\text{prime}\). However, if, for example, \(y=\text{prime}\) and \(x=4\), \(xy=4y \neq \text{prime}\). Not sufficient.
(1)+(2) From (1), we know that \(x=1\) or \(x=-1\). From (2), we know that \(x > 0\). Combining these, we get \(x=1\). Therefore, \(xy=1*\text{prime}=\text{prime}\). Sufficient.
Answer: C
If \(x\) and \(y\) are integers is \(xy\) a prime number?
Firstly that only positive integers can be prime numbers and that 1 is not a prime number.
Secondly, for the product of two integers \(x\) and \(y\) to be a prime number, either one of them must be \(1\) and the other a prime number, for example (1, 2), (1, 3), (1, 5), (1, 7), ..., or one of them must be \(-1\) and the other \(-\text{prime}\), for example (-1, -2), (-1, -3), (-1, -5), (-1 ,-7), ...
(1) \(x^2=1\).
Either \(x=1\) or \(x=-1\). Not sufficient.
(2) \(y\) is a prime number and \(x\) is a positive number but not a prime number.
If \(y=\text{prime}\) and \(x=1\), then \(xy=y=\text{prime}\). However, if, for example, \(y=\text{prime}\) and \(x=4\), \(xy=4y \neq \text{prime}\). Not sufficient.
(1)+(2) From (1), we know that \(x=1\) or \(x=-1\). From (2), we know that \(x > 0\). Combining these, we get \(x=1\). Therefore, \(xy=1*\text{prime}=\text{prime}\). Sufficient.
Answer: C
General Discussion
