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16 Sep 2014, 00:20
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D
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Difficulty:
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Question Stats:
65% (01:25) correct
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wrong
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Is the product of three integers \(p\), \(q\), and \(r\) even?
(1) \((p - 1)(r + 1)\) is odd.
(2) \((q - r)^2\) is odd.
(1) \((p - 1)(r + 1)\) is odd.
(2) \((q - r)^2\) is odd.
16 Sep 2014, 00:20
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Official Solution:
Is the product of three integers \(p\), \(q\), and \(r\) even?
For a product of integers to be even, at least one of the integers must be even. For a product of integers to be odd, all of the integers must be odd. Hence, \(pqr\) will be even if at least one of the unknowns is even and will be odd if all of the unknowns are odd.
(1) \((p - 1)(r + 1)\) is odd.
According to the above, \((p - 1)(r + 1)\) to be odd, both \(p - 1\) and \(r + 1\) must be odd. Therefore, both \(p\) and \(r\) are even. Sufficient.
(2) \((q - r)^2\) is odd.
For the square of an integer to be odd, the integer itself must be odd. Therefore \(q - r\) is odd. Next, for a difference of two integers to be odd, one of the two must be even and another must be odd. One even integer is sufficient to make the product \(pqr\) even. Sufficient.
Answer: D
Is the product of three integers \(p\), \(q\), and \(r\) even?
For a product of integers to be even, at least one of the integers must be even. For a product of integers to be odd, all of the integers must be odd. Hence, \(pqr\) will be even if at least one of the unknowns is even and will be odd if all of the unknowns are odd.
(1) \((p - 1)(r + 1)\) is odd.
According to the above, \((p - 1)(r + 1)\) to be odd, both \(p - 1\) and \(r + 1\) must be odd. Therefore, both \(p\) and \(r\) are even. Sufficient.
(2) \((q - r)^2\) is odd.
For the square of an integer to be odd, the integer itself must be odd. Therefore \(q - r\) is odd. Next, for a difference of two integers to be odd, one of the two must be even and another must be odd. One even integer is sufficient to make the product \(pqr\) even. Sufficient.
Answer: D
General Discussion
28 Feb 2023, 08:53
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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.
