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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. \(\frac{3x^2}{2} + 1\)

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.

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

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29 Nov 2012, 15:09

1

This post received KUDOS

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
_________________

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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