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# How many even integers N exist such that x < N < x + 10?

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How many even integers N exist such that x < N < x + 10?  [#permalink]

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27 Jan 2017, 03:35
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85% (hard)

Question Stats:

43% (01:51) correct 57% (01:58) wrong based on 316 sessions

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How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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27 Jan 2017, 04:28
1
lmuenzel wrote:
How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

(1) if x= 0.1 then total no. of even integers (0.1< N < 10.1) = 5
if x=2 then total number of even integers (2<N<12) = 4
Not suff

(2) if x= non integer
then N= 5 always
sufficient

Ans B
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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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27 Jan 2017, 08:01
lmuenzel wrote:
How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

This question test the concept of number line. 'Not odd' means any other integers, fraction etc except ODD number. Now just plug the given equation those 'not odd' values.

Ans: B
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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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27 Jan 2017, 09:23
lmuenzel wrote:
How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

Check similar questions to practice:
how-many-integers-n-are-there-such-that-r-n-s-166396.html
how-many-integers-are-there-such-that-v-n-w-129065.html

Hope it helps.
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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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27 Jan 2017, 22:06
x < N < X+10

Stmt-1:
if x is not odd then what else can it be? it can be even. but do you see the second stmt as well, oh yes! x can be fraction too

So you have two cases:
when x is even say x=0 then you have 0 < 2,4,6,8 < 10 - in total 4 even numbers
when x is fraction say x=0.5 then you have 0.5 < 2,4,6,8,10 < 10.5 - in total 5 even numbers

hence statement-1 is insufficient.

stmt-2:
when x is fraction say x=0.5 then you have 0.5 < 2,4,6,8,10 < 10.5 - in total 5 even numbers
when x is fraction say x=1.5 then you have 1.5 < 2,4,6,8,10 < 11.5 - in total 5 even numbers
when x is fraction say x=2.5 then you have 2.5 < 4,6,8,10,12 < 12.5 - in total 5 even numbers

Sufficient.
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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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23 Dec 2017, 04:41
Easy deal by number picking. Questions asks the number of even integers between x and (x+10)

1. If x is an even integer
Ex: x = 0 ----> 0< 2,4,6,8 < 10 = 4

If x is not an integer but is even
Ex: x = 0.5 ----> 0.5 < 2,4,6,8,10 < 10.5 = 5

Clearly not sufficient.

2. If x is not an integer
Ex: x = 0.5 -----> 0.5 < 2,4,6,8,10 < 10.5 = 5

Sufficient.

Option B

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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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06 Aug 2018, 22:45
1
why x can't be negative?
not odd doesn't mean x can't be negative.
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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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22 Nov 2018, 17:15
lmuenzel wrote:
How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

$$x\,\,\, < \,\,\,N\,\,{\text{even}}\,\,\, < \,\,\,x + 10$$

$$? = \# N$$

$$\left( 1 \right)\,\,x \ne {\text{odd}}\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,x{\text{ = 0}}\,\,\,\, \Rightarrow \,\,\,\,? = 4\,\,\,\,\,\,\,\left[ {2,4,6\,\,{\text{and}}\,\,8} \right] \hfill \\ \,{\text{Take}}\,\,x = 0.1\,\,\,\, \Rightarrow \,\,\,\,? = 5\,\,\,\,\,\,\,\left[ {2,4,6,8\,\,{\text{and}}\,\,10} \right] \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,x \ne \operatorname{int} \,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,x < \left\langle x \right\rangle \leqslant N \leqslant \left\langle {x + 9} \right\rangle < x + 10$$

$$\Rightarrow \,\,\,\,\left\{ \begin{gathered} \,\left\langle x \right\rangle \,\,{\text{odd}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {x + 9} \right\rangle \,\,{\text{even}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}} = {\text{5}}\,\,\,\left[ {\left\langle {x + j} \right\rangle :j \in \left\{ {1,3,5,7,9} \right\}} \right] \hfill \\ \,\left\langle x \right\rangle \,\,{\text{even}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {x + 9} \right\rangle \,\,{\text{odd}}\,\,\,\, \Rightarrow \,\,\,\,{\text{?}} = {\text{5}}\,\,\,\left[ {\left\langle {x + j} \right\rangle :j \in \left\{ {0,2,4,6,8} \right\}} \right] \hfill \\ \end{gathered} \right.\,\,\,\, \Rightarrow \,\,\,\,\,? = 5$$

$$\left( * \right)\,\,\left\langle r \right\rangle \,\, = \,\,{\text{smallest}}\,\,{\text{integer}}\,\,{\text{greater}}\,\,{\text{than}}\,\,r$$

The correct answer is therefore (B).

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: How many even integers N exist such that x < N < x + 10?  [#permalink]

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22 Nov 2018, 17:27
nightvision wrote:
why x can't be negative?
not odd doesn't mean x can't be negative.

Hi, nightvision !

When statement (1) is considered, a BIFURCATION is all you need to prove insufficiency.

(You COULD bifurcate (1) using negative numbers, of course.)

Statement (2) is a lot more interesting (and its corresponding rigorous treatment much harder).

My solution (posted above) deals with all possible scenarios.

If you cannot feel comfortable with it, let me give you a suggestion:

Substitute x by 0.1 (so that <x> will be 1) and do all the reasoning.
Afterwards, do exactly the same with 1.1 (in this case <x> will be 2).

If you prefer negative numbers, LoL, try x = -1.1 (in which case <x> equals -1).
Afterwards, do exactly the same with x = -0.1 (in this case <x> equals 0).

I believe after (at most) all 4 substitutions mentioned, you will REALLY understand all my arguments!

Regards and success in your studies,
Fabio.
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How many even integers N exist such that x < N < x + 10?  [#permalink]

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21 Dec 2018, 03:55
lmuenzel wrote:
How many even integers N exist such that x < N < x + 10?

(1) x is not odd

(2) x is not an integer

Veritas Prep OFFICIAL EXPLANATION

There are various ways to approach a problem like this one.

Students who are familiar with the principles of evenly spaced sets (so-called “Fence Post Problems”) can immediately infer that the number of even integers in this set will depend on whether the excluded endpoints x and x+10 are themselves even or not.

For these students, it is important to be careful when interpreting statement (1). When reading that x is not odd, recognize that two possibilities still remain – even, of course, but also non-integer. One case excludes the endpoints, but the other does not, so the number of elements in the set will change by 1 between these two cases (it will be either 4 or 5, respectively). Statement (2), however, guarantees that the endpoints are not excluded, so the number of even integers will simply be Range/Spacing = 10/2 = 5.

Other students may wish to pick a few possible values for x to reach the same conclusion. In that case, it’s important to consider different types of numbers. What types? Since the additional statements reference whether x is odd and whether it’s an integer, it would be wise to choose at least one even integer, one odd integer, and one non-integer.

For x is even, we will always find four even integers N satisfying x < N < x+10. E.g. x=10, 10 < N < 20, N could be 12, 14, 16, or 18.

For x is odd, we will always find five even integers N satisfying x < N < x+10. E.g. x=1. 1 < N < 11, N could be 2, 4, 6, 8, or 10.

For x is a non-integer, we will once again always find five even integers N satisfying x < N < x+10. E.g. x=-0.5. -0.5 < N < 9.5, N could be 0, 2, 4, 6, or 8.

Using either approach, statement (1) is not sufficient to answer the question, but statement (2) is. The answer is B.
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How many even integers N exist such that x < N < x + 10?   [#permalink] 21 Dec 2018, 03:55
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