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

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

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

(1) x is not odd

(2) x is not an integer

I have another question, so we ignore statement (1) and acknowledge that it is insufficient, but where does statement (2) or the question tell us that x can't be odd?

For example it doesn't say that 3 < 4, 6, 8, 10, 12 < 13 does it?

Or does the question imply that the numbers are sequential?
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Re: How many even integers N exist such that x < N < x + 10? [#permalink]
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

$$x < N < x + 10$$

Scenarios:
$$1.1 < N < 10.1$$ gives us 5 even integers
$$2 < N < 12$$ gives us 4 even integers
$$1 < N < 11$$ gives us 5 integers

(1) If x = not odd, we can still get four or five even integers, depending on if x = non-integer or x = even. INSUFFICIENT.

(2) If x = not an integer, we will have 5 even integers. SUFFICIENT.

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