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If \(x\) and \(y\) are positive integers, is \(x\) odd?
(1) \(x - 2y\) is a prime number
(2) \(x + 2y\) is a prime number
(1) \(x - 2y\) is a prime number
(2) \(x + 2y\) is a prime number
04 Jul 2019, 05:45
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Official Solution:
If \(x\) and \(y\) are positive integers, is \(x\) odd?
(1) \(x - 2y\) is a prime number.
If \(x - 2y=even \ prime\), that is if \(x - 2y=2\), then \(x = even+2y=even+even=even\) (consider \(x=4\) and \(y=1\) as an example). However, if \(x - 2y=odd \ prime\), then \(x = odd+2y=odd+even=odd\) (consider \(x=5\) and \(y=1\) as an example). Not sufficient.
(2) \(x + 2y\) is a prime number.
Given that \(x\) and \(y\) are positive integers, it's impossible for \(x + 2y\) to equal 2, the only even prime number. This is because \(x + 2y = \{minimum \ 1\} + \{ minimum \ 2\} = \{minimum \ 3\} > 2\). Therefore, \(x + 2y\) must equal an \(odd \ prime\). In this case, \(x =odd \ prime-2y=odd-even=odd\). Sufficient.
Answer: B
If \(x\) and \(y\) are positive integers, is \(x\) odd?
(1) \(x - 2y\) is a prime number.
If \(x - 2y=even \ prime\), that is if \(x - 2y=2\), then \(x = even+2y=even+even=even\) (consider \(x=4\) and \(y=1\) as an example). However, if \(x - 2y=odd \ prime\), then \(x = odd+2y=odd+even=odd\) (consider \(x=5\) and \(y=1\) as an example). Not sufficient.
(2) \(x + 2y\) is a prime number.
Given that \(x\) and \(y\) are positive integers, it's impossible for \(x + 2y\) to equal 2, the only even prime number. This is because \(x + 2y = \{minimum \ 1\} + \{ minimum \ 2\} = \{minimum \ 3\} > 2\). Therefore, \(x + 2y\) must equal an \(odd \ prime\). In this case, \(x =odd \ prime-2y=odd-even=odd\). Sufficient.
Answer: B
General Discussion
19 May 2023, 13:34
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
