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# If x is a positive number

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If x is a positive number [#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.
[Reveal] Spoiler: OA
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Re: If x is a positive number [#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 [#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 [#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 [#permalink]

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30 Aug 2012, 05:18
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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 [#permalink]

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30 Aug 2012, 19:50
great job
great way to solve this problem..

i didnt reach the answer as you did

ds has been troubling me like anything
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Re: If x is a positive number [#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 [#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 NO 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 [#permalink]

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Re: If x is a positive number [#permalink]

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24 Aug 2016, 13:05
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Re: If x is a positive number [#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 [#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 [#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 [#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 [#permalink]

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19 Apr 2017, 17:56
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)
Re: If x is a positive number   [#permalink] 19 Apr 2017, 17:56
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