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# If x is even integer, which of the following must be an odd

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If x is even integer, which of the following must be an odd [#permalink]

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25 Jul 2012, 07:57
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If x is even integer, which of the following must be an odd integer?

A. $$\frac{3x}{2}$$
B. $$\frac{3x}{2} + 1$$
C. $$3x^2$$
D. $$\frac{3x^2}{2}$$
E. $$\frac{3x^2}{2} + 1$$
[Reveal] Spoiler: OA

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Last edited by Bunuel on 25 Jul 2012, 08:03, edited 1 time in total.
Edited the question.
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Re: If x is even integer, which of the following must be an odd [#permalink]

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29 Nov 2012, 15:09
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I narrowed this question down to B and E. Based on the rules alone why couldn't it be B?

(3x/2)+1

if x is even then we have an even + odd which would be odd?

thanks for the clarification.

Because the theory is important but also to reach the answer through the most efficient way.

$$x=2$$(as statement says) $$OR x=4$$ (thanks this we know for instance that A is not always true)

$$\frac{6}{2}$$$$= 3+ 1 = 4$$ $$OR 7$$ (is not always true: one time even one time odd). That's it Same for the other answer choices. You can obtain E in 30 seconds
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Re: If x is even integer, which of the following must be an odd [#permalink]

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30 Nov 2012, 03:54
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I narrowed this question down to B and E. Based on the rules alone why couldn't it be B?

(3x/2)+1

if x is even then we have an even + odd which would be odd?

thanks for the clarification.

That's not correct.

Given that $$x$$ is even, thus $$x=2k$$ for some integer $$k$$. Substitute in option B:

$$\frac{3x}{2}+1=\frac{3*2k}{2}+1=3k+1$$ --> if $$k=odd$$ then $$3k+1=odd+1=even$$ but if $$k=even$$ then $$3k+1=even+1=odd$$.

As you can see $$\frac{3x}{2}+1$$ can be even (for example if x is 2, 6, 10, ...) as well as odd (for example if x is 4, 8, 12, ...).

Hope it's clear.
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Re: If x is even integer, which of the following must be an odd [#permalink]

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25 Jul 2012, 08:05
Stiv wrote:
If x is even integer, which of the following must be an odd integer?

A. $$\frac{3x}{2}$$
B. $$\frac{3x}{2} + 1$$
C. $$3x^2$$
D. $$\frac{3x^2}{2}$$
E. $$\frac{3x^2}{2} + 1$$

One can spot right away that if $$x$$ is any even number then $$x^2$$ is a multiple of 4, which makes $$\frac{x^2}{2}$$ an even number and therefore $$\frac{3x^2}{2}+1=3*even+1=even+1=odd$$.

If you don't notice this, then one also do in another way. Let $$x=2k$$, for some integer k, then:

A. $$\frac{3x}{2}=\frac{3*2k}{2}=3k$$ --> if $$k=odd$$ then $$3k=odd$$ but if $$k=even$$ then $$3k=even$$. Discard;

B. $$\frac{3x}{2}+1=\frac{3*2k}{2}+1=3k+1$$ --> if $$k=odd$$ then $$3k+1=odd+1=even$$ but if $$k=even$$ then $$3k+1=even+1=odd$$. Discard;

C. $$3x^2$$ --> easiest one as $$x=even$$ then $$3x^2=even$$, so this option is never odd. Discard;

D. $$\frac{3x^2}{2}=\frac{3*4k^2}{2}=6k^2=even$$, so this option is never odd. Discard;

E. $$\frac{3x^2}{2}+1=\frac{3*4k^2}{2}=6k^2+1=even+1=odd$$, thus this option is always odd.

Similar question to practice: if-a-and-b-are-positive-integers-such-that-a-b-and-a-b-are-88108.html

Hope it helps.
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Re: If x is even integer, which of the following must be an odd [#permalink]

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25 Jul 2012, 21:26
Hi,

Since, x is even we can assume x = 2 or x = 4,
such that x/2 is both odd/even as per the value of x.

Check each option with these values. We get the answer when both x=2, 4 gives an odd value.

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Re: If x is even integer, which of the following must be an odd [#permalink]

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29 Nov 2012, 15:04
I narrowed this question down to B and E. Based on the rules alone why couldn't it be B?

(3x/2)+1

if x is even then we have an even + odd which would be odd?

thanks for the clarification.
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Re: If x is even integer, which of the following must be an odd [#permalink]

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29 Nov 2012, 16:06
thanks.

certainly i'm trying to answer "odd/even" questions in the most efficient manner possible.

what i would have done on the real CAT is narrowed it down to B and E, then like you let x = 2 or 4 and plugged in to see.

i kind of got tripped up. typically when we multiply a integer by an even we ALWAYS get an even but (3/2) is frac, therefore multiplying it by an even may or may not make it even?
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Re: If x is even integer, which of the following must be an odd [#permalink]

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30 Nov 2012, 07:40
thank you for the help.
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Re: If x is even integer, which of the following must be an odd [#permalink]

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13 Apr 2017, 01:57
Option E

x is an Even inetegr. Odd: O, Even: E & Fraction: F

A. $$\frac{3x}{2}$$ = $$\frac{3*E}{2}$$ = $$3*E$$ $$or$$ $$3*O$$= $$E$$ $$or$$ $$O$$
B. $$\frac{3x}{2}+1$$ = $$\frac{3*E}{2} + 1$$ = $$3*E + 1$$ $$or$$ $$3*O + 1$$= $$O$$ $$or$$ $$E$$
C. $$3x^2$$ = $$3E^2$$ = $$3*E$$ = $$E$$
D. $$\frac{3x^2}{2}$$ = $$\frac{3E^2}{2}$$ = $$3*E$$ = $$E$$
E. $$\frac{3x^2}{2}+1$$ = $$\frac{3E^2}{2}+1$$ = $$3*E + 1$$ = $$E + 1$$ = $$O$$
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Re: If x is even integer, which of the following must be an odd   [#permalink] 13 Apr 2017, 01:57
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