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# If n is an integer, is n positive?

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Intern
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If n is an integer, is n positive?  [#permalink]

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Updated on: 09 Nov 2015, 16:49
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95% (hard)

Question Stats:

49% (01:36) correct 51% (01:38) wrong based on 577 sessions

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If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Originally posted by jcmsolis2015 on 09 Nov 2015, 16:38.
Last edited by ENGRTOMBA2018 on 09 Nov 2015, 16:49, edited 2 times in total.
Formatted the question and updated the topic.
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Re: If n is an integer, is n positive?  [#permalink]

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09 Nov 2015, 19:38
6
6
jcmsolis2015 wrote:
If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Target question: Is n positive?

Given: In is an integer

Statement 1:(2n+1)/(n+1) is an integer
Rewrite as: (n+1+n)/(n+1) is an integer
Rewrite as: (n+1)/(n+1) + n/(n+1) is an integer
Or... 1 + n/(n+1) is an integer
This means that n/(n+1) is an integer
How can this be? How can some number divided by a number that's 1 greater be an integer?
This ONLY WORKS if n = 0, so statement 1 is really telling us that n = 0
In other words, n is NOT positive
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n=−n
Add n to both sides to get: 2n = 0
Divide both sides by 2 to get: n = 0
So, n is NOT positive
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: If n is an integer, is n positive?  [#permalink]

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09 Nov 2015, 16:48
3
3
jcmsolis2015 wrote:
If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Given that n is an integer, is n > 0?

Per statement 2, n=-n ---> 2n=0 ---> n=0. Answer to "is n > 0" = NO. Sufficient.

Per statement 1, (2n+1)/(n+1) = integer ---> n =0 and n =-2 satisfy the conditions (out of some other negative values, no need to check any more). No positive integer values of n will satisfy this given condition. Thus you get a "no" for "is n > 0" for both n =0 and n = -2 , making this statement sufficient as well.

D is thus the correct answer.
##### General Discussion
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If n is an integer, is n positive?  [#permalink]

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10 Nov 2015, 00:18
Engr2012 wrote:
jcmsolis2015 wrote:
If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Given that n is an integer, is n > 0?

Per statement 2, n=-n ---> 2n=0 ---> n=0. Answer to "is n > 0" = NO. Sufficient.

Per statement 1, (2n+1)/(n+1) = integer ---> n =0 and n =-2 satisfy the conditions (out of some other negative values, no need to check any more). No positive integer values of n will satisfy this given condition. Thus you get a "no" for "is n > 0" for both n =0 and n = -2 , making this statement sufficient as well.

D is thus the correct answer.

I must disagree with the reasoning on statement 1. Just because you gave 2 examples in which the statement holds true, that does not mean the statement does not hold true for any positive values of N. The message from this is: the only scenario in which you can use two examples to prove that a statement is sufficient or not, is when you (relating to this question) have a value of N for which the answer is YES and a value for N for which the answer is NO. Proving that the statement is insufficient. I guessed B on this question.

-update-

On statement 1 I reasoned that the only way for it to be true would be 2n=n meaning that n should equal 0.
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Re: If n is an integer, is n positive?  [#permalink]

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10 Nov 2015, 08:33
2
Martijn wrote:
I must disagree with the reasoning on statement 1. Just because you gave 2 examples in which the statement holds true, that does not mean the statement does not hold true for any positive values of N. The message from this is: the only scenario in which you can use two examples to prove that a statement is sufficient or not, is when you (relating to this question) have a value of N for which the answer is YES and a value for N for which the answer is NO. Proving that the statement is insufficient. I guessed B on this question.
.

Martijn is correct; plugging in numbers typically works best when you suspect that the statement is NOT SUFFICIENT. In these cases, all you need to do is find values that yield different (conflicting) answers to the target question.

Conversely, if the statement is SUFFICIENT, then plugging in values will only HINT at whether or not the statement is sufficient, but you won't be able to make any definitive conclusions.

For example, let's say we have the following target question: If n is a positive integer, is (2^n) - 1 prime?
Let's say statement 1 says: n is a prime number:

Now let's plug in some prime values of n:
If n = 2, then (2^n) - 1 = 2² - 1 = 3, and 3 IS prime
If n = 3, then case (2^n) - 1 = 2³ - 1 = 7, and 7 IS prime
If n = 5, then (2^n) - 1 = 2⁵ - 1 = 31, and 31 IS prime
At this point, it certainly APPEARS that statement guarantees that (2^n) - 1 is prime? Let’s try one more prime value of n.
If n = 7, then (2^n) - 1 = 2⁷ - 1 = 127, and 127 IS prime

So, can we be 100% certain that statement 1 is sufficient? No. The truth of the matter is that statement 1 is NOT SUFFICIENT. To see why, let’s examine the possibility that n = 11
If n = 11, then (2^n) - 1 = (2^11) - 1 = 2047, and 2047 is NOT prime

Here’s a different example:
Target question: Is x > 0?
Let's say statement 1 says: 5x > 4x

Now let's plug in some values of x that satisfy the condition that 5x > 4x.
x = 3, in which case x > 0
x = 0.5, in which case x > 0
x = 15, in which case x > 0
x = 1000, in which case x > 0

Once again, it APPEARS that statement 1 provides sufficient information to answer the target question. Can we be 100% certain? No. Perhaps we didn't plug in the right numbers (as was the case in the first example). Perhaps there's a number that we could have plugged in such that x < 0

If we want to be 100% certain that a statement is SUFFICIENT, we'll need to use a technique other than plugging in.
Here, we can take 5x > 4x, and subtract 4x from both sides to get x > 0 VOILA - we can now answer the target question with absolute certainty.
So, statement 1 is SUFFICIENT.

TAKEAWAY: Plugging in numbers is best suited for situations in which you suspect that the statement is not sufficient. In these situations, plugging in values can yield results that are 100% conclusive. Conversely, in situations in which the statement is sufficient, plugging in values can STRONGLY HINT at sufficiency, but the results are not 100% conclusive.

For more on this, you can watch our free video titled “Choosing Good Numbers: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1102 or you can read an article I wrote: http://www.gmatprepnow.com/articles/dat ... lug-values

Cheers,
Brent
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Re: If n is an integer, is n positive?  [#permalink]

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11 Nov 2015, 00:23
GMATPrepNow wrote:
Martijn wrote:
I must disagree with the reasoning on statement 1. Just because you gave 2 examples in which the statement holds true, that does not mean the statement does not hold true for any positive values of N. The message from this is: the only scenario in which you can use two examples to prove that a statement is sufficient or not, is when you (relating to this question) have a value of N for which the answer is YES and a value for N for which the answer is NO. Proving that the statement is insufficient. I guessed B on this question.
.

Martijn is correct; plugging in numbers typically works best when you suspect that the statement is NOT SUFFICIENT. In these cases, all you need to do is find values that yield different (conflicting) answers to the target question.

Conversely, if the statement is SUFFICIENT, then plugging in values will only HINT at whether or not the statement is sufficient, but you won't be able to make any definitive conclusions.

For example, let's say we have the following target question: If n is a positive integer, is (2^n) - 1 prime?
Let's say statement 1 says: n is a prime number:

Now let's plug in some prime values of n:
If n = 2, then (2^n) - 1 = 2² - 1 = 3, and 3 IS prime
If n = 3, then case (2^n) - 1 = 2³ - 1 = 7, and 7 IS prime
If n = 5, then (2^n) - 1 = 2⁵ - 1 = 31, and 31 IS prime
At this point, it certainly APPEARS that statement guarantees that (2^n) - 1 is prime? Let’s try one more prime value of n.
If n = 7, then (2^n) - 1 = 2⁷ - 1 = 127, and 127 IS prime

So, can we be 100% certain that statement 1 is sufficient? No. The truth of the matter is that statement 1 is NOT SUFFICIENT. To see why, let’s examine the possibility that n = 11
If n = 11, then (2^n) - 1 = (2^11) - 1 = 2047, and 2047 is NOT prime

Here’s a different example:
Target question: Is x > 0?
Let's say statement 1 says: 5x > 4x

Now let's plug in some values of x that satisfy the condition that 5x > 4x.
x = 3, in which case x > 0
x = 0.5, in which case x > 0
x = 15, in which case x > 0
x = 1000, in which case x > 0

Once again, it APPEARS that statement 1 provides sufficient information to answer the target question. Can we be 100% certain? No. Perhaps we didn't plug in the right numbers (as was the case in the first example). Perhaps there's a number that we could have plugged in such that x < 0

If we want to be 100% certain that a statement is SUFFICIENT, we'll need to use a technique other than plugging in.
Here, we can take 5x > 4x, and subtract 4x from both sides to get x > 0 VOILA - we can now answer the target question with absolute certainty.
So, statement 1 is SUFFICIENT.

TAKEAWAY: Plugging in numbers is best suited for situations in which you suspect that the statement is not sufficient. In these situations, plugging in values can yield results that are 100% conclusive. Conversely, in situations in which the statement is sufficient, plugging in values can STRONGLY HINT at sufficiency, but the results are not 100% conclusive.

For more on this, you can watch our free video titled “Choosing Good Numbers: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1102 or you can read an article I wrote: http://www.gmatprepnow.com/articles/dat ... lug-values

Cheers,
Brent

Hi Brent,
I have quick question regarding your example is x>0. If I use x=-2 in the statement 5x>4x, so -10<-8 so X won't be smaller than zero. it should be insufficient. However, through simple algebra you proved that X>0. I'm confused. what did I miss?

Thanks
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Re: If n is an integer, is n positive?  [#permalink]

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11 Nov 2015, 07:03
Mo2men wrote:

Hi Brent,
I have quick question regarding your example is x>0. If I use x=-2 in the statement 5x>4x, so -10<-8 so X won't be smaller than zero. it should be insufficient. However, through simple algebra you proved that X>0. I'm confused. what did I miss?

Thanks

Plugging in x = -2 does not satisfy the given inequality 5x > 4x
We get: (5)(-2) > (4)(-2), which evaluates to -10 > -8, HOWEVER -10 is not greater than -8

Cheers,
Brent
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Re: If n is an integer, is n positive?  [#permalink]

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11 Nov 2015, 07:27
GMATPrepNow wrote:
Mo2men wrote:

Hi Brent,
I have quick question regarding your example is x>0. If I use x=-2 in the statement 5x>4x, so -10<-8 so X won't be smaller than zero. it should be insufficient. However, through simple algebra you proved that X>0. I'm confused. what did I miss?

Thanks

Plugging in x = -2 does not satisfy the given inequality 5x > 4x
We get: (5)(-2) > (4)(-2), which evaluates to -10 > -8, HOWEVER -10 is not greater than -8

Cheers,
Brent

Shouldn't I reverse the sign because of multiplication with negative number to be -10<-8. I will happy if you spot my mistake in my understanding.

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Re: If n is an integer, is n positive?  [#permalink]

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15 Nov 2015, 00:58
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

There is one variable (n) in the original condition and 2 equations in the given conditions, so there is high chance (D) is the answer.
For condition 1, n=0,-2, so the answer is 'no' and is sufficient.
For condition 2, 2n=0, n=0, so the answer is 'no' and is sufficient, making the answer (D).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: If n is an integer, is n positive?  [#permalink]

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08 Jan 2017, 21:21
GMATPrepNow wrote:
jcmsolis2015 wrote:
If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Target question: Is n positive?

Given: In is an integer

Statement 1:(2n+1)/(n+1) is an integer
Rewrite as: (n+1+n)/(n+1) is an integer
Rewrite as: (n+1)/(n+1) + n/(n+1) is an integer
Or... 1 + n/(n+1) is an integer
This means that n/(n+1) is an integer
How can this be? How can some number divided by a number that's 1 greater be an integer?
This ONLY WORKS if n = 0, so statement 1 is really telling us that n = 0
In other words, n is NOT positive
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n=−n
Add n to both sides to get: 2n = 0
Divide both sides by 2 to get: n = 0
So, n is NOT positive
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent

Hi Brent,

We can plug -2 in statement 1 and get result as 3 which is an integer. We can plug 0 and get result as 1 which is also an integer. There are 2 values of n(-2 and 0) that can give integer as result. Doesn't this imply that Statement 1 is insufficient to answer whether n is positive?
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Re: If n is an integer, is n positive?  [#permalink]

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06 Jun 2017, 21:02
getisb wrote:
GMATPrepNow wrote:
jcmsolis2015 wrote:
If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Target question: Is n positive?

Given: In is an integer

Statement 1:(2n+1)/(n+1) is an integer
Rewrite as: (n+1+n)/(n+1) is an integer
Rewrite as: (n+1)/(n+1) + n/(n+1) is an integer
Or... 1 + n/(n+1) is an integer
This means that n/(n+1) is an integer
How can this be? How can some number divided by a number that's 1 greater be an integer?
This ONLY WORKS if n = 0, so statement 1 is really telling us that n = 0
In other words, n is NOT positive
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n=−n
Add n to both sides to get: 2n = 0
Divide both sides by 2 to get: n = 0
So, n is NOT positive
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent

Hi Brent,

We can plug -2 in statement 1 and get result as 3 which is an integer. We can plug 0 and get result as 1 which is also an integer. There are 2 values of n(-2 and 0) that can give integer as result. Doesn't this imply that Statement 1 is insufficient to answer whether n is positive?

Are we missing something in our understanding ?
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If n is an integer, is n positive?  [#permalink]

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06 Jun 2017, 21:07
Ah, I got it now. In both the cases (with values of n as 0 or -2), we are able to get an integer value as an output and thus, are able to answer the main question (is N positive ?) definitively as in both the cases, we are getting a NO (n is NOT positive with values of 0 and -2) and thus, statement 1 is sufficient.
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Re: If n is an integer, is n positive?  [#permalink]

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01 Nov 2017, 09:39
Statement 1:
(2n + 1)/(n+1) is integer
Let (2n + 1)/(n+1) = k
rearranging gives => n = (k-1)/(k-2)
Now note that k-1 and k-2 are co-primes, for n to be integer, (k-1)/(k-2) must be integer
For that, k - 1 = k -2 , which is definitely not possible,
or
k - 1 = 0 (to make n as integer) => k =1 => n = 0

so n must be zero and not positive, sufficient to answer the question

Statement 2:
n = -n, only n = 0, this is valid , sufficient

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Re: If n is an integer, is n positive?  [#permalink]

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01 Nov 2017, 18:41
(1) (2n+1)/(n+1) is an integer. We have (2n+1)/ (n+1) = (n+1 +n)/(n+1) = 1 + n/ (n+1) is an integer
=> n/(n+1) is an integer <=> n = 0 => n isn't positive => sufficient

(2): n=−n <=> 2n = 0 <=> n =0 => n isn't positive => sufficient

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Re: If n is an integer, is n positive?  [#permalink]

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25 Sep 2018, 09:25
GMATPrepNow wrote:
jcmsolis2015 wrote:
If n is an integer, is n positive?

1) (2n+1)/(n+1) is an integer

2) n=−n

Target question: Is n positive?

Given: In is an integer

Statement 1:(2n+1)/(n+1) is an integer
Rewrite as: (n+1+n)/(n+1) is an integer
Rewrite as: (n+1)/(n+1) + n/(n+1) is an integer
Or... 1 + n/(n+1) is an integer
This means that n/(n+1) is an integer
How can this be? How can some number divided by a number that's 1 greater be an integer?
This ONLY WORKS if n = 0, so statement 1 is really telling us that n = 0
In other words, n is NOT positive
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n=−n
Add n to both sides to get: 2n = 0
Divide both sides by 2 to get: n = 0
So, n is NOT positive
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent

Hi Brent,

In the above statement 1, can n be -2? Because in the 3rd step above in which 1+ n/(n+1), if we replace n with -2, the answer will be 2. So technically n can be 0 or -2? However, the statement still prove to be sufficient.
Re: If n is an integer, is n positive? &nbs [#permalink] 25 Sep 2018, 09:25
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