Last visit was: 23 Apr 2024, 23:44 It is currently 23 Apr 2024, 23:44

GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# If n is an integer, is n even?

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 92883
Own Kudos [?]: 618653 [21]
Given Kudos: 81563
Math Expert
Joined: 02 Sep 2009
Posts: 92883
Own Kudos [?]: 618653 [22]
Given Kudos: 81563
General Discussion
Intern
Joined: 22 Nov 2012
Affiliations: CA, SAP FICO
Posts: 32
Own Kudos [?]: 24 [0]
Given Kudos: 51
Location: India
Concentration: Finance, Sustainability
GMAT 1: 620 Q42 V33
GMAT 2: 720 Q47 V41
GPA: 3.2
WE:Corporate Finance (Energy and Utilities)
Math Expert
Joined: 02 Sep 2009
Posts: 92883
Own Kudos [?]: 618653 [1]
Given Kudos: 81563
Re: If n is an integer, is n even? [#permalink]
1
Bookmarks
X017in wrote:
Bunuel wrote:
SOLUTION:

If n is an integer, is n even?

(1) n^2 - 1 is an odd integer --> $$n^2-1=odd$$ --> $$n^2=odd+1=even$$. Now, since $$n$$ is an integer, then in order $$n^2$$ to be even $$n$$ must be even. Sufficient.
Notice that if we were not told that $$n$$ is an integer, then $$n$$ could be some irrational number (square root of an even number), for example $$\sqrt{2}$$, so not an even integer.

(2) 3n + 4 is an even integer --> $$3n + 4=even$$ --> $$3n=even-4=even$$. The same here, since $$n$$ is an integer, then in order $$3n$$ to be even $$n$$ must be even. Sufficient.
Notice that if we were not told that $$n$$ is an integer, then $$n$$ could be some fraction, for example $$\frac{2}{3}$$, so not an even integer.

Since n is a integer, can we not try with n as 0?

Yes, n can be 0 but 0 is even too.
Tutor
Joined: 20 Aug 2015
Posts: 350
Own Kudos [?]: 1392 [1]
Given Kudos: 10
Location: India
GMAT 1: 760 Q50 V44
Re: If n is an integer, is n even? [#permalink]
1
Kudos
Bunuel wrote:
If n is an integer, is n even?

(1) n^2 - 1 is an odd integer.
(2) 3n + 4 is an even integer.

Given: n is an integer
Required: is n even?

Statement 1: $$n^2$$ - 1 is an odd integer
$$n^2$$ - 1 = (n-1)(n+1) = odd.
This means both n-1 and n+1 are odd
Odd*Odd = Odd
Odd*Even = Even
Even*Even = Even

n-1, n, n+1 are three consecutive integers.
Since we know that both n-1 and n+1 are odd
Hence n has to be even.

SUFFICIENT

Statement 2: 3n + 4 is an even integer
Even + Even = Even
Even + Odd = Odd
Odd + Odd + Odd

Since 3n+4 = even and 4 is an even integer.
Hence 3n = even. Therefore n = even
SUFFICIENT

Option D
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 18753
Own Kudos [?]: 22042 [2]
Given Kudos: 283
Location: United States (CA)
Re: If n is an integer, is n even? [#permalink]
2
Kudos
Quote:
If n is an integer, is n even?

(1) n^2 - 1 is an odd integer.
(2) 3n + 4 is an even integer.

We need to determine whether integer n is even. Let's review four facts about even and odd integers: 1) An integer and its square are either both even or both odd. 2) The sum (or difference) between an even integer and an odd integer is always odd. 3) The sum of two even integers (or two odd integers) is always even. 4) If the product of two integers is even, at least one of them must be even.

Statement One Alone:

(n^2) - 1 is an odd integer.

Since (n^2) - 1 is an odd integer, we know that n^2 must be even and thus n must be even. Statement one is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

3n + 4 is an even integer.

Since 3n + even = even integer, we know that 3n must be even, and since 3 is odd, n must be even. Statement two is sufficient to answer the question.

GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6821
Own Kudos [?]: 29899 [1]
Given Kudos: 799
Re: If n is an integer, is n even? [#permalink]
1
Kudos
Top Contributor
Bunuel wrote:
If n is an integer, is n even?

(1) n^2 - 1 is an odd integer.
(2) 3n + 4 is an even integer.

Some important rules:
1. ODD +/- ODD = EVEN
2. EVEN +/- ODD = ODD
3. EVEN +/- EVEN = EVEN

4. (ODD)(ODD) = ODD
5. (ODD)(EVEN) = EVEN
6. (EVEN)(EVEN) = EVEN

Target question: Is integer n EVEN?

Statement 1: n² - 1 is an odd integer
n² - 1 = (n + 1)(n - 1)
So, statement 1 is telling us that (n + 1)(n - 1) = ODD
From rule #4 (above), we can conclude that BOTH (n + 1) and (n - 1) are ODD
If (n + 1) is ODD, then n must be EVEN (since 1 is ODD, we can apply rule #2 to conclude that n is EVEN)
If (n - 1) is ODD, then n must be EVEN (by rule #2 )
So, the answer to the target question is YES, n is even
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 3n + 4 is an even integer
In other words, (3n + EVEN) is EVEN
From rule #3, we can conclude that 3n is EVEN

Since 3 is odd, we can write: (ODD)(n) = EVEN
From rule #5, we can conclude that n is EVEN
So, the answer to the target question is YES, n is even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
Intern
Joined: 04 Feb 2020
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 1
Re: If n is an integer, is n even? [#permalink]
Hello, i have a question. What if n=0.

Then (1) 0^2 - 1 is an odd integer. But n is not an even number.

For (2) 3x0 + 4 is an even integer. But n is not an even number.

Should the answer be E or there is something i am missing?
Math Expert
Joined: 02 Sep 2009
Posts: 92883
Own Kudos [?]: 618653 [0]
Given Kudos: 81563
Re: If n is an integer, is n even? [#permalink]
diegoecheverria wrote:
Hello, i have a question. What if n=0.

Then (1) 0^2 - 1 is an odd integer. But n is not an even number.

For (2) 3x0 + 4 is an even integer. But n is not an even number.

Should the answer be E or there is something i am missing?

Zero is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.
Non-Human User
Joined: 09 Sep 2013
Posts: 32636
Own Kudos [?]: 821 [0]
Given Kudos: 0
Re: If n is an integer, is n even? [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Re: If n is an integer, is n even? [#permalink]
Moderator:
Math Expert
92883 posts