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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x is an even integer, which of the following must be an odd integer

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Intern  Joined: 28 Aug 2007
Posts: 9
If x is an even integer, which of the following must be an odd integer  [#permalink]

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Question Stats: 74% (01:28) correct 26% (01:06) wrong based on 794 sessions

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If x is an 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) $$\frac{3x^2}{2} + 1$$

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

Originally posted by slsu on 09 Oct 2007, 23:08.
Last edited by Bunuel on 10 Oct 2019, 04:42, edited 4 times in total.
Edited the question and added the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 65014
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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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 $$\frac{3x^2}{2}+1=3*even+1=even+1=odd$$.

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: even-and-odd-gmatprep-88108.html

Hope it helps.
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Director  Joined: 09 Aug 2006
Posts: 528
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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1
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.
Manager  Joined: 10 Aug 2007
Posts: 59
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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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.

Intern  Joined: 28 Aug 2007
Posts: 9
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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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.

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?
Intern  Joined: 18 May 2007
Posts: 8
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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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.
Intern  Joined: 28 Aug 2007
Posts: 9
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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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
Manager  Joined: 07 Sep 2007
Posts: 82
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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3
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
Intern  Joined: 11 Nov 2009
Posts: 1
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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1
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.
Manager  Status: One last try =,=
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Posts: 122
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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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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Manager  Joined: 01 Dec 2011
Posts: 62
Re: If x is an even integer, which of the following must be an odd integer  [#permalink]

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

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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

Since x is even, x^2 must be even also. Furthermore, x^2 must be a multiple of 4 since it will have at least two factors of 2. Therefore, 3x^2/2 is even and since even + 1 = odd, 3x^2/2 + 1 will always be odd.

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

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_________________ Re: If x is an even integer which of the following must be an   [#permalink] 10 Oct 2019, 04:41

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