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If x is an even integer, which of the following must be an [#permalink]
 10 Oct 2007, 00:08
Question Stats:
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22% (00:44) wrong based on 1 sessions
If x is an even integer, which of the following must be an odd integer? (A) 3x/2 (B) [3x/2] + 1 (C) 3x^2 (D) [3x^2]/2 (E) [3x^2/2] + 1 I know that this is easiest (and safest) by plugging in numbers, but I'm curious to know if there are any other number theory rules I could use, besides these standard rules: O*E = E E*E = E O*O = O Or maybe I'm just thinking too hard into the question
Last edited by Bunuel on 26 Jan 2012, 06:28, edited 2 times in total.
Edited the question and added the OA
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Re: If x is an even integer, which of the following must be. [#permalink]
10 Oct 2007, 00:19
slsu wrote: This is from a GMATPrep Exam:
If x is an even integer, which of the following must be an odd integer?
(A) 3x/2
(B) [3x/2] + 1
(C) 3x^2
(D) [3x^2]/2
(E) [3x^2/2] + 1
I know that this is easiest (and safest) by plugging in numbers, but I'm curious to know if there are any other number theory rules I could use, besides these standard rules:
O*E = E
E*E = E
O*O = O
Or maybe I'm just thinking too hard into the question
Well,
E+O = O
O+O = E
etc.
For this question, answer should be E.
3x^2/2 + 1
if x is even, x^2 is always even. If x is an even integer then x^2/2 is always even. E*O = E, which means 3*x^2/2 is always even. E+O = O which means 3x^2/2 + 1 is odd.
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I think that with those 3 rules you have your solution already 
x is E (even) and therefore x^2 is E.
3x^2 is E, as it is O*E
Therefore, 3x^2/2 will be E. (since x is integer, x^2 > 2)
If you add 1 to an E number, you will always get odd.
Hence, E is the answer.
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after15 wrote: I think that with those 3 rules you have your solution already x is E (even) and therefore x^2 is E. 3x^2 is E, as it is O*E Therefore, 3x^2/2 will be E. (since x is integer, x^2 > 2) If you add 1 to an E number, you will always get odd. Hence, E is the answer. Ah ha! That's the ticket - I forgot that if x = Even, then x^2 = Even.
I didn't quite understand the statement:
since x is integer, x^2 > 2. Did you mean x^2 > x?
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slsu wrote: Ok, I understand that we can eliminate A, C, and D, but how can you distinguish between B & E, if the same ODD/EVEN properties apply to both?
B) 3x/2 + 1
[O*E]/[E] + O
E*E + O
E+O = O
E) 3x^2/2 + 1
[O*E]/[E] + O
E*E + O
E+O = O
I agree that the answer is E, but I see why B is confusing. I concluded in E simply b/c of process of elimination. I can disprove (b) by plugging in x=2 ==>4==>even.
I guess that's the right way to approach it.
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Re: If x is an even integer, which of the following must be. [#permalink]
10 Oct 2007, 11:18
GK_Gmat wrote: slsu wrote: This is from a GMATPrep Exam:
If x is an even integer, which of the following must be an odd integer?
(A) 3x/2
(B) [3x/2] + 1
(C) 3x^2
(D) [3x^2]/2
(E) [3x^2/2] + 1
I know that this is easiest (and safest) by plugging in numbers, but I'm curious to know if there are any other number theory rules I could use, besides these standard rules:
O*E = E
E*E = E
O*O = O
Or maybe I'm just thinking too hard into the question Well, E+O = O O+O = E etc. For this question, answer should be E. 3x^2/2 + 1 if x is even, x^2 is always even. If x is an even integer then x^2/2 is always even. E*O = E, which means 3*x^2/2 is always even. E+O = O which means 3x^2/2 + 1 is odd. Ok, then, is it safe to suppose then, that every EVEN number raised to a power, divided by that same base (2), must be EVEN:
2^2 = 4/2 = 2 (E)
2^3 = 8/2 = 4 (E)
2^4 = 16/2 = 8 (E)
This is opposed to an EVEN number divided by an EVEN number, which can either result in an EVEN or ODD number:
2/2 = 1
4/2 = 2
6/2 = 3
8/2 = 4
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Ady wrote: It is tricky. You'd only notice it without trying answers if you happened to notice that that x/2 is even for all even values except x=2 or x =-2
This is not true.
x = 6 for instance.
The easiest approach to this answer is to count the "minimum" even prime factors (aka 2s).
If we know X is even, we have at least one 2 as a factor of X.
If we have X^2, we double all those factors.
Thus, X^2/2 is guaranteed to be even.
Even + 1 = Odd
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Re: If x is an even integer, which of the following must be...? [#permalink]
29 May 2010, 17:58
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The reason why (3X^2/2) + 1 is even is consider just X^2/2 part.
1) First X^2/2 can be written as X.X/2
2) X is even.
3) X/2 can be Even or Odd
3) That means X.X/2 is Even*Even OR Odd number
4) This number is always Even.
5) 3*Even number is Even.
6) Even number + 1 is ODD.
There we arrive at the answer. Simple as that.
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Re: If x is an even integer, which of the following must be...? [#permalink]
30 Aug 2010, 01:25
One way to choose between B and C: n is even ---> n = 2k with k is an integer From B: 3x/2 + 1 = 3k + 1 --> the result can be either odd or even, depending on k From E: 3x^2/2 + 1 = 6k^2 + 1 ---> always odd
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JingChan wrote: Ady wrote: It is tricky. You'd only notice it without trying answers if you happened to notice that that x/2 is even for all even values except x=2 or x =-2 This is not true. x = 6 for instance. The easiest approach to this answer is to count the "minimum" even prime factors (aka 2s). If we know X is even, we have at least one 2 as a factor of X. If we have X^2, we double all those factors. Thus, X^2/2 is guaranteed to be even. Even + 1 = Odd +1 for the precise explanation In option B, given that x is 6 (2 x 3=6) for example, the 2 can be cancelled out so that 3x/2 results in 9, which is an odd integer. 9+1=10, which is even. However, once there are more than one 2s, the fraction will always result in an even number and finally add up to an odd number.
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Re: This is from a GMATPrep Exam: If x is an even integer, [#permalink]
26 Jan 2012, 06:27
slsu wrote: If x is an even integer, which of the following must be an odd integer?
A. 3x/2
B. 3x/2+1
C. 3x^2
D. 3x^2/2
E. 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: even-and-odd-gmatprep-88108.htmlHope it helps.
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Re: If x is an even integer, which of the following must be an [#permalink]
26 Jan 2012, 07:02
A bit tricky but got the answer as E
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Re: If x is an even integer, which of the following must be an
[#permalink]
26 Jan 2012, 07:02
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