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# If x is even, which of the following could be odd ?

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Joined: 01 Sep 2010
Posts: 3397
If x is even, which of the following could be odd ?  [#permalink]

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04 Apr 2018, 16:25
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Difficulty:

35% (medium)

Question Stats:

70% (01:40) correct 30% (01:30) wrong based on 72 sessions

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If x is even, which of the following could be odd ?

A. $$\frac{(x^2 + x)(x + 2)}{4}$$

B. $$x^3 - 3x^2 + 2x$$

C. $$x^2 - 4x + 6$$

D. $$\frac{x^3 + x^2}{4}$$

E. $$\frac{(x - 2)(x^2 + 2x)}{4}$$

Kudos for the right answer and explanation

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Joined: 30 Mar 2017
Posts: 127
GMAT 1: 200 Q1 V1
Re: If x is even, which of the following could be odd ?  [#permalink]

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04 Apr 2018, 17:36
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Answer choice B and C will always be even, so can eliminate those. The numerator of choice A, D, and E will always be even, but when divided by 4, may give an odd number. What is one example of an even number that, when divided by 4, will result in an odd number? 12. Choice D immediately sticks out because 2^3=8 and 2^2=4.

For a more systematic approach, can do this...

Since we've narrowed down the answers to A, D, or E, and all 3 have 4 in the denominator, that means the numerator must be of the form 4*odd so that the 4 in the numerator and the 4 in the denominator cancel out and we're left with the odd number.

Let's examine each numerator.

Choice A
$$(x^2+x)(x+2)=x(x+1)(x+2)$$
If x=2, the even factors wont cancel out. x=0 wont work and picking other even values for x will just introduce more even factors. So this can't be odd.

Choice D
$$x^3+x^2=x^2(x+1)$$
If x=2, then the even factors cancel out and we're left with an odd number.

Choice E
$$(x-2)(x^2+2x)=x(x-2)(x+2)$$
All factors are even.
Re: If x is even, which of the following could be odd ?   [#permalink] 04 Apr 2018, 17:36
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