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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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New post 25 Jul 2012, 06:57
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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, 07:03, edited 1 time in total.
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Re: If x is even integer, which of the following must be an odd [#permalink]

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buymovieposters wrote:
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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buymovieposters wrote:
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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New post 25 Jul 2012, 07:05
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 \(\frac{3x^2}{2}+1=3*even+1=even+1=odd\).

Answer: E.

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.

Answer: E.

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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New post 25 Jul 2012, 20:26
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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New post 29 Nov 2012, 14:04
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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New post 29 Nov 2012, 15:06
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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New post 30 Nov 2012, 06:40
thank you for the help.
Re: If x is even integer, which of the following must be an odd   [#permalink] 30 Nov 2012, 06:40
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