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# If x is a positive number, is x an even integer?

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If x is a positive number, is x an even integer?  [#permalink]

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26 Aug 2012, 06:24
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If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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26 Aug 2012, 06:42
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If x is a positive number, is x an even integer?

Notice that we are not told that x is an integer.

(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

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Re: If x is a positive number, is x an even integer?  [#permalink]

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26 Aug 2012, 09:13
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syog wrote:
If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

Neither (1) nor (2) alone is sufficient.

(1) and (2) together:

From (1) $$3x=2k,$$ where $$k$$ is a positive integer.
From (2) $$5x=2m,$$ where $$m$$ is some positive integer. Necessarily $$\frac{2k}{3}=\frac{2m}{5}$$ from which $$5k=3m.$$
$$k$$ and $$m$$ being integers, necessarily $$k$$ must be a multiple of 3 (because 5 is not divisible by 3), so $$k=3a$$ for some positive integer $$a.$$
It follows that $$x=\frac{2k}{3}=2a$$ so $$x$$ is even.

Sufficient.

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Re: If x is a positive number, is x an even integer?  [#permalink]

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30 Aug 2012, 04:53
Bunuel wrote:
If x is a positive number, is x an even integer?

Notice that we are not told that x is an integer.

(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

My answer to this question was D, both sufficient, however my assumption was that in GMAT number and integer are interchangible words, but as i see i was wrong. Bunuel could you please remind what word was interchangible with word integer?
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Re: If x is a positive number, is x an even integer?  [#permalink]

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30 Aug 2012, 05:18
ziko wrote:
Bunuel wrote:
If x is a positive number, is x an even integer?

Notice that we are not told that x is an integer.

(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

My answer to this question was D, both sufficient, however my assumption was that in GMAT number and integer are interchangible words, but as i see i was wrong. Bunuel could you please remind what word was interchangible with word integer?

I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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02 Sep 2013, 08:23
Bunuel wrote:
ziko wrote:
Bunuel wrote:
If x is a positive number, is x an even integer?

Notice that we are not told that x is an integer.

(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

My answer to this question was D, both sufficient, however my assumption was that in GMAT number and integer are interchangible words, but as i see i was wrong. Bunuel could you please remind what word was interchangible with word integer?

I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.

I am a little confused.
1) 3x is an even integer.
Let x=4/3; then 3x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

2) 5x is an even integer.
Let x=4/5; then 5x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

(1+2): 15x is an integer.
Let x=4/15; then 15x=4. But in this case x is not an even integer. Hence INSUFFICIENT.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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02 Sep 2013, 08:42
domfrancondumas wrote:
Bunuel wrote:
ziko wrote:
If x is a positive number, is x an even integer?

Notice that we are not told that x is an integer.

(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

My answer to this question was D, both sufficient, however my assumption was that in GMAT number and integer are interchangible words, but as i see i was wrong. Bunuel could you please remind what word was interchangible with word integer?

I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.

I am a little confused.
1) 3x is an even integer.
Let x=4/3; then 3x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

2) 5x is an even integer.
Let x=4/5; then 5x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

(1+2): 15x is an integer.
Let x=4/15; then 15x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

Notice that x cannot be 4/15, because in this case 3x=12/15 which is NOT an even integer and 5x=20/15 which is also NOT an even integer, so in this case both statements are violated.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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10 Jan 2017, 07:31
If it's given that "5 is a factor of x". Is it correct to assume that x is an integer? If yes, how and why? Please explain. Also explain if the answer to my question is No. Consider a case wherein x is 5/2. Isn't 5 a factor of x? Please address, thanks a lot in advance
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Re: If x is a positive number, is x an even integer?  [#permalink]

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10 Jan 2017, 10:12
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kooks123 wrote:
If it's given that "5 is a factor of x". Is it correct to assume that x is an integer? If yes, how and why? Please explain. Also explain if the answer to my question is No. Consider a case wherein x is 5/2. Isn't 5 a factor of x? Please address, thanks a lot in advance

On the GMAT when we are told that $$a$$ is divisible by $$b$$ (or which is the same: "$$a$$ is multiple of $$b$$", or "$$b$$ is a factor of $$a$$"), we can say that:
1. $$a$$ is an integer;
2. $$b$$ is an integer;
3. $$\frac{a}{b}=integer$$.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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11 Jan 2017, 03:48
x is positive Integer means we need to think only in positive fractions and integers.

(1) 3x is an even integer.

Let x= 2.....3x=6..........x is even integer

Let x=2/3....3x=2.........x is NOT even integer

Insufficient

(2) 5x is an even integer.

Let x= 2........5x=10..........x is even integer

Let x=2/5.......5x=2...........x is NOT even integer

Insufficient

Combining 1 +2 ,

There is not fraction could be multiplied simultaneously to 3 & 5 and give EVEN INTEGER. We left with that x=EVEN INTEGER to give make both 3x & 5x even integers.

As 3 & 5 are odd numbers then x needs to be EVEN Integer to make both statements Even integers.

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Re: If x is a positive number, is x an even integer?  [#permalink]

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11 Jan 2017, 12:01
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Hi Ellipse,

Whenever you combine two statements in a DS question, the best thing to do instead of plugging in values is to use one statement into the other or use a mathematical operation between the two statements. The mathematical operation(s) that you choose to perform must lead to your target question. In this case the target question is 'Is x an even integer'?

Statement 1 : 3x is an even integer

Here x can either be an even integer such as 2, 4, 6.... or it can be a fraction such as 2/3, 4/3..... So x can be an even integer or a fraction. Insufficient.

Statement 2 : 5x is an even integer

Here again x can be an even integer such as 2, 4, 6.... or it can be a fraction such as 2/5, 4/5.... So x again can either be an even integer or a fraction. Insufficient.

Now instead of recycling the values it makes sense to use a mathematical operation (or mathematical operations) between the two statements.

Let us multiply the first statement by 2, since 3x = even integer ; 3x * 2 = even integer * 2 -----> 6x = even integer

Statement 2 says that 5x is an even integer and by multiplying statement 1 by 2 we have 6x to be an even integer. Subtracting the two we get

6x - 5x = even integer - even integer -----> x = even integer. Sufficient.

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Re: If x is a positive number, is x an even integer?  [#permalink]

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19 Apr 2017, 17:56
1
Ellipse wrote:
If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

Statement 1:

3 (2/3) could be an even number- the questin says x is a positive number...not necessarily an integer

Insufficient

Statement 2

5(2/5) btw again here were are using a counterexample to establish that x does not necessarily have to be an integer
10/5 =2

Insufficient

Statement 1 and 2:

Statement 1 and Statement 2

Knowing from statement 1 and 2 that 3x and 5x are even integers we can use solve this question using algebra instead of imagining numbers and testing various cases- an even integer minus an even integer is always an even integer so

5x-3x=2x
5(4)-3(4)=2(4)
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Re: If x is a positive number, is x an even integer?  [#permalink]

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22 Aug 2017, 12:36
Bunuel wrote:
ziko wrote:
Bunuel wrote:
If x is a positive number, is x an even integer?

Notice that we are not told that x is an integer.

(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

My answer to this question was D, both sufficient, however my assumption was that in GMAT number and integer are interchangible words, but as i see i was wrong. Bunuel could you please remind what word was interchangible with word integer?

I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.

Dear, Natural numbers are not non-negative numbers as 0 is a non-negative Integer but 0 is not natural number. Natural numbers are positive integers.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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22 Aug 2017, 13:04
HolaMaven wrote:
Bunuel wrote:
I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.

Dear, Natural numbers are not non-negative numbers as 0 is a non-negative Integer but 0 is not natural number. Natural numbers are positive integers.

There are two definitions as mentioned in my post:
https://en.wikipedia.org/wiki/Natural_number
http://mathworld.wolfram.com/NaturalNumber.html
http://www.mathopenref.com/counting-number.html
https://www.wolframalpha.com/input/?i=natural+numbers
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Re: If x is a positive number, is x an even integer?  [#permalink]

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22 Aug 2017, 13:18
Bunuel wrote:
HolaMaven wrote:
Bunuel wrote:
I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.

Dear, Natural numbers are not non-negative numbers as 0 is a non-negative Integer but 0 is not natural number. Natural numbers are positive integers.

There are two definitions as mentioned in my post:
https://en.wikipedia.org/wiki/Natural_number
http://mathworld.wolfram.com/NaturalNumber.html
http://www.mathopenref.com/counting-number.html
https://www.wolframalpha.com/input/?i=natural+numbers

Thank you for your enlightenment on the topic. Do the definitions differ according to the area of usage? Like in Number Theory we have Natural Numbers starting from 1 and in Set Theory or Computer Science Natural Numbers includes 0.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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22 Aug 2017, 13:20
1) 3x is even with x=2 and x= 2/3 , hence A,D ruled out
2) 5x is even with x= 2 and x= 2/5 B ruled out

3x = 2,4,6,8,10,12.....
x = 2/3, 4/3, 6/3,8/3,.....

5x = 2,4,6,8,10....
x= 2/5,4/5,6/5,8/5,10/5

taking common from above series x=2,4,6....and so on...hence even.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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22 Aug 2017, 13:27
1
HolaMaven wrote:
Bunuel wrote:
HolaMaven wrote:
Dear, Natural numbers are not non-negative numbers as 0 is a non-negative Integer but 0 is not natural number. Natural numbers are positive integers.

There are two definitions as mentioned in my post:
https://en.wikipedia.org/wiki/Natural_number
http://mathworld.wolfram.com/NaturalNumber.html
http://www.mathopenref.com/counting-number.html
https://www.wolframalpha.com/input/?i=natural+numbers

Thank you for your enlightenment on the topic. Do the definitions differ according to the area of usage? Like in Number Theory we have Natural Numbers starting from 1 and in Set Theory or Computer Science Natural Numbers includes 0.

Yes, there are different definitions:
"The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (OEIS A000027) or to the set of nonnegative integers 0, 1, 2, 3, ... (OEIS A001477; e.g., Bourbaki 1968, Halmos 1974). Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 element P."

Luckily we don't have to worry about this at all. Not a single official GMAT question uses a term "natural number".

GMAT questions use: integer, positive integer, negative integer, non-negative integer.
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Re: If x is a positive number, is x an even integer?  [#permalink]

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22 Aug 2017, 13:50
Bunuel
good day Bunuel,

let in case if a ques come in gmat then whether i have to include 0 in natural number or not. as u mentioned above that gmat has not asked natural numbers specifically in its questions.
till today i was knowing that 0 is a whole number. and is not a natural number.

P.S. thanks
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Re: If x is a positive number, is x an even integer?  [#permalink]

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10 Dec 2017, 15:24
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Hi All,

We're told that X is a POSITIVE number (meaning that X > 0). We're asked if X is an EVEN INTEGER. This is a YES/NO question. This question can be solved by TESTing VALUES.

1) 3X is an even integer.

IF....
X = 2/3, then 3X = 2 and the answer to the question is NO.
X = 2, then 3X = 6 and the answer to the question is YES.
Fact 1 is INSUFFICIENT

2) 5X is an even integer.
X = 2/5, then 5X = 2 and the answer to the question is NO.
X = 2, then 5X = 10 and the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, the various 'fractional' answers that fit one of the Facts will NOT fit the other (since multiplying the same fraction by 3 and by 5 won't 'cancel out' the denominator and lead to an integer result for both calculations). Thus, the only possible values for X must be INTEGERS. In addition, since 3 and 5 are both ODD integers, for the 3X and 5X to both be EVEN integers, the X MUST be EVEN.
Combined, SUFFICIENT

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