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If positive integers \(q\) and \(r\) are both even, which of the following must be odd?

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If positive integers \(q\) and \(r\) are both even, which of the following must be odd?

by M7MBA » Wed Sep 09, 2020 6:41 am

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If positive integers \(q\) and \(r\) are both even, which of the following must be odd?

A. \(q - r\)
B. \(\dfrac{q}{r}\)
C. \(\dfrac{q}{r} + 1\)
D. \(qr - 1\)
E. \(q(r - 1)\)

Answer: D

Source: Princeton Review
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Source: — Problem Solving |
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Re: If positive integers \(q\) and \(r\) are both even, which of the following must be odd?

by Nirav Umaretiya » Sat Sep 12, 2020 10:56 pm
You can solve this one by using the method of elimination. In that, you need to try to eliminate the option by all means and if you can't eliminate, that one is the answer.

A) q- r
so, even - even = even
i.e. 4 - 2 = 2 (Eliminate)

B) \(\frac{q}{r}\)
so \(\frac{even}{even}\) = even
i.e. \(\frac{4}{2}\) = 2 (Eliminate)

C) \(\frac{q}{r}\) - 1
so, \(\frac{even}{even}\) - 1 = Odd - 1 = Even
i.e. \(\frac{6}{2}\) = 3-1 = 2 (Eliminate)

D) qr - 1
so, (even)(even) - 1 = (even) - 1 = Odd
i.e (2)(2) - 1 = 4-1 = 3 = Odd (Seems correct, no reason to eliminate)

E) q (r-1)
so, even (even-1) = even
i.e. 2 (4-1) = 2(3) = 6 (Eliminate)

Now select D as all else can be eliminated.
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