Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 23 Aug 2011
Posts: 78

If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
23 Aug 2012, 01:17
5
This post received KUDOS
21
This post was BOOKMARKED
Question Stats:
47% (01:06) correct 53% (01:12) wrong based on 656 sessions
HideShow timer Statistics
If a and n are integers and n > 1, is n odd? (1) a^(n–1) > a^n (2) a^n > a^(3n) Please provide me pointers to any similar questions, also if possible Can some one explain steps to tackle such problems with least amount of time.For eg. how to select the values for plugging etc. THanks
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Whatever one does in life is a repetition of what one has done several times in one's life! If my post was worth it, then i deserve kudos



Director
Status: Tutor  BrushMyQuant
Joined: 05 Apr 2011
Posts: 606
Location: India
Concentration: Finance, Marketing
GPA: 3
WE: Information Technology (Computer Software)

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
23 Aug 2012, 02:11
5
This post was BOOKMARKED
Am not srue as to how to substitute values and slove this. Am writing the algebric solution to this problems If a and n are integers and n > 1, is n odd? (1) a^(n–1) > a^n a^(n1)  a^n > 0 a^(n1) (1 a) >0 Either both a^(n1) and (1a) are > 0 or both are < 0 taking the case in which both are > 0 a^(n1) >0 1a> 0 => a<1 and a^(n1) > 0 so, only possible solution to this is a < 0 and n1 = even => n = odd taking the case in which both are < 0 a^(n1) < 0 and 1a < 0 => a > 1 There is no solution to thsi equation as a^(n1) for a > 1 will always be > 0 So, SUFFICIENT! (2) a^n > a^(3n) a^n  a^3n >0 a^n(1a^2n) > 0 Either both a^n and (1a^2n) are > 0 or both are < 0 taking the case when both are > 0 a^n > 0 , 1 a^2n > 0 => a^2n < 1 and a^n > 0 This equation is true for negative a and negative even values of n, But n > 1 so, NO solution! taking the case when both are < 0 a^n < 0 , 1 a^2n < 0 => a^2n > 1 and a^n < 0 This equation is true for negative values of a and odd values of n, So n is odd, SUFFICIENT Answer is D
_________________
Ankit
Check my Tutoring Site > Brush My Quant
GMAT Quant Tutor How to start GMAT preparations? How to Improve Quant Score? Gmatclub Topic Tags Check out my GMAT debrief
How to Solve : Statistics  Reflection of a line  Remainder Problems  Inequalities



GMAT Tutor
Joined: 24 Jun 2008
Posts: 1346

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
23 Aug 2012, 18:35
conty911 wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n) If you factor, as in the post above, you can avoid cases if you arrange things differently. Rewriting Statement 1, noticing that a cannot be 1, we can factor out the smaller power, so can factor out a^(n1) : 0 > a^n  a^(n1) 0 > a^(n1) (a  1) Now one term on the right side must be positive, and the other must be negative. But if a is a positive integer, that can't possibly be true  both terms will be positive. So a must be negative, and (a1) must be the negative term, and a^(n1) must be the positive term. Since a is negative, that means the exponent must be even, so n1 is even and n is odd. Similarly for Statement 2, notice a cannot be 1, 0 or 1, and we have 0 > a^(3n)  a^n 0 > a^n [a^(2n)  1] and again, one term must be positive, the other negative. But a^(2n) has an even exponent, so it must be positive and so a^(2n)  1 is the positive term (using the fact that a cannot be 0, 1 or 1). That makes a^n the negative term. But if a^n is negative, a must be negative, and n must be odd. I don't think you need to do the problem algebraically, however. You can just imagine how different numbers would 'behave' in the inequalities given. So if you know that n is a positive integer greater than 1, and you know a^(n1) > a^n then you might be able to see that a must be negative, since this inequality won't ever be true if a > 1. And if we know the base is negative, and we can see that one exponent is even and the other is odd (since the exponents are just consecutive integers), we know that one side of the inequality will become positive, and the other will remain negative. Clearly the larger side needs to be positive, which makes n1 even, and n odd. Similarly for the other Statement, the inequality a^n > a^(3n) simply cannot be true if a > 1. Since 1, 0 or 1 don't work in this inequality, a must be 2 or smaller. But then if n is even, it wouldn't matter if a is negative or positive, since even exponents just 'erase' our negative signs. And we know it must matter, since the inequality won't work if we have positive bases on either side. So the inequality will only work if n is odd so that the left and right sides both remain negative.
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com



Director
Joined: 22 Mar 2011
Posts: 608
WE: Science (Education)

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
04 Oct 2012, 03:06
5
This post received KUDOS
1
This post was BOOKMARKED
conty911 wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n)
Please provide me pointers to any similar questions, also if possible
Can some one explain steps to tackle such problems with least amount of time.For eg. how to select the values for plugging etc. THanks From the conditions in both statements, we can deduce that \(a\) cannot be neither \(0\) nor \(1.\) A little algebraic manipulation can help in understanding the inequalities and also in choosing values for \(a\). Usually it is easier to compare a product of numbers to \(0\), so it is worth rearranging the inequalities such that there is a comparison to \(0\) involved. (1) \(a^{n1}>a^n\) can be rewritten as \(a^{n1}a^n>0\) or \(a^{n1}(1a)>0\). If \(a>1\), doesn't matter if \(n\) is even or odd, the given inequality cannot hold. If \(a<0,\) then necessarily \(n\) must be odd. Sufficient. (2)\(a^n>a^{3n}\) can be rewritten as \(a^na^{3n}\) or \(a^n(1a^{2n})>0\). Since \(a\) cannot be neither \(0\) nor \(1,\) \(\,\,1a^{2n}\) is certainly negative. (Now \(a\) cannot be \(1\) either.) Then \(a^n\) is negative if \(a\) is negative and \(n\) is odd, and this is the only way the given inequality can hold. If \(a\) is positive, doesn't matter the value of \(n\), the inequality cannot hold. Sufficient. Answer D.
_________________
PhD in Applied Mathematics Love GMAT Quant questions and running.
Last edited by EvaJager on 05 Oct 2012, 01:08, edited 3 times in total.



Manager
Joined: 10 May 2012
Posts: 53
GMAT Date: 09102012

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
05 Oct 2012, 00:54
conty911 wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n)
Please provide me pointers to any similar questions, also if possible
Can some one explain steps to tackle such problems with least amount of time.For eg. how to select the values for plugging etc. THanks Consider substituting numbers: statement 1 : To get stmt 1 true n=2 (even) and a = 2 (+ve numbers will never satisfy condition) 2^1>2^2 = 2>4 (No) n=3 (odd) and a = 2 2^2>2^3 = 4>8 (Yes) Sufficient Statement 2 : To get stmt 2 true n=2 (even) and a = 2 (+ve numbers will never satisfy condition) 2^2>2^6 = 4>64 (No) n=3 (odd) and a = 2 2^3>2^9 = 8>512 (Yes) Sufficient Ans D



Manager
Status: Pushing Hard
Affiliations: GNGO2, SSCRB
Joined: 30 Sep 2012
Posts: 85
Location: India
Concentration: Finance, Entrepreneurship
GPA: 3.33
WE: Analyst (Health Care)

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
26 Apr 2013, 22:01
IanStewart wrote: conty911 wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n) If you factor, as in the post above, you can avoid cases if you arrange things differently. Rewriting Statement 1, noticing that a cannot be 1, we can factor out the smaller power, so can factor out a^(n1) : 0 > a^n  a^(n1) 0 > a^(n1) (a  1) Now one term on the right side must be positive, and the other must be negative. But if a is a positive integer, that can't possibly be true  both terms will be positive. So a must be negative, and (a1) must be the negative term, and a^(n1) must be the positive term. Since a is negative, that means the exponent must be even, so n1 is even and n is odd. Similarly for Statement 2, notice a cannot be 1, 0 or 1, and we have 0 > a^(3n)  a^n 0 > a^n [a^(2n)  1] and again, one term must be positive, the other negative. But a^(2n) has an even exponent, so it must be positive and so a^(2n)  1 is the positive term (using the fact that a cannot be 0, 1 or 1). That makes a^n the negative term. But if a^n is negative, a must be negative, and n must be odd. I don't think you need to do the problem algebraically, however. You can just imagine how different numbers would 'behave' in the inequalities given. So if you know that n is a positive integer greater than 1, and you know a^(n1) > a^n then you might be able to see that a must be negative, since this inequality won't ever be true if a > 1. And if we know the base is negative, and we can see that one exponent is even and the other is odd (since the exponents are just consecutive integers), we know that one side of the inequality will become positive, and the other will remain negative. Clearly the larger side needs to be positive, which makes n1 even, and n odd. Similarly for the other Statement, the inequality a^n > a^(3n) simply cannot be true if a > 1. Since 1, 0 or 1 don't work in this inequality, a must be 2 or smaller. But then if n is even, it wouldn't matter if a is negative or positive, since even exponents just 'erase' our negative signs. And we know it must matter, since the inequality won't work if we have positive bases on either side. So the inequality will only work if n is odd so that the left and right sides both remain negative.  Hi Ian, I like the way you solved this Question, but I really can't understand the 2nd stmt.. explanation. Pls explain it to me. Thanks !!
_________________
If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.



Manager
Status: Pushing Hard
Affiliations: GNGO2, SSCRB
Joined: 30 Sep 2012
Posts: 85
Location: India
Concentration: Finance, Entrepreneurship
GPA: 3.33
WE: Analyst (Health Care)

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
26 Apr 2013, 22:04
Bunuel wrote: Archit143 wrote: If a and n are integers and n > 1, is n odd?
(1) (a)^n–1 > a^n (2) a^n > a^(3n)  Hi Bunuel.... I have read the explanation provided by Ian ,... but I'm unable to understand the explanation for 2nd statement ,, & I think Ian is no longer a regular visitor to the GMAT Club ..... Can You pls... Make me understand this question ....& solution Pls,,,,,, Thanks !!
_________________
If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.



MBA Section Director
Status: Back to work...
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 5061
Location: India
City: Pune
GPA: 3.4
WE: Business Development (Manufacturing)

Re: Is N Odd ?? [#permalink]
Show Tags
29 Apr 2013, 12:31
manishuol wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n) S1) a^(n–1) > a^n This can happen in two cases. If a is negative (e.g. 2^2 > 2^3) of if a is fraction (e.g. (1/2)^2 > (1/2)^3. Since we are told that both a and n are integers, later case is not possible. So we can conclude that a is negative. Now to hold the inequality a^(n–1) > a^n true when a being negative, n1 must be even and hence n must be odd. Sufficient S2) In the same way as S1 this choice is also sufficient Ans = D Regards, Narenn
_________________
Chances of Getting Admitted After an Interview [Data Crunch]
Must Read Forum Topics Before You Kick Off Your MBA Application
New GMAT Club Decision Tracker  Real Time Decision Updates



Senior Manager
Joined: 21 Jan 2010
Posts: 323

Re: Is N Odd ?? [#permalink]
Show Tags
29 Apr 2013, 17:01
If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n)
Well , i feel a bit intimidated by equation questions so it took me 2:25 to solve this one. But if you keep the initial hesitation aside , this can be solved in less than 2 min.
From 1) a^n1 > a ^n. From given conditions ie n> 1 and a and n being integers , this is only possible if a is negative and n1 is even integer. so n is an odd integer. From 2) a^n > a^3n. Again only possible if a is negative and n has to be odd.
So D is the answer.
Last edited by Bluelagoon on 29 Apr 2013, 17:09, edited 1 time in total.



Retired Moderator
Joined: 05 Jul 2006
Posts: 1741

Re: Is N Odd ?? [#permalink]
Show Tags
29 Apr 2013, 17:06
If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n)
from 1
a^(n–1) > a^n thus , a^(n1)  a^n > 0 , thus a^n (a^1  1) > 0 i.e. a^n [ (1/a)  1] >0 and n>1 and 1/a is a fraction therefore a^n has to be ve , this is only possible when a is ve integer and n is odd .....suff
from 2
a^n > a^(3n), thus , a^n  a^ 3n > 0 thus a^n ( 1 a^2n) > 0 and since a,n are integers and n>1 and since (1a^2n) is definitely ve therefore a^n is ve and n is odd ....suff
D



Intern
Joined: 20 Jun 2011
Posts: 45

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
29 May 2013, 12:53
EvaJager wrote: conty911 wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n)
Please provide me pointers to any similar questions, also if possible
Can some one explain steps to tackle such problems with least amount of time.For eg. how to select the values for plugging etc. THanks \(a^{n1}a^n>0\) or \(a^{n1}(1a)>0\). Can someone please explain me stepwise how the factorization is done? Thanks.



Math Expert
Joined: 02 Sep 2009
Posts: 44400

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
29 May 2013, 12:57



Intern
Joined: 20 Jun 2011
Posts: 45

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
29 May 2013, 13:06
Bunuel wrote: We are factoring our \(a^{n1}\) from \(a^{n1}a^n\) to get \(a^{n1}(1a)\):
\(a^{n1}(1a)=a^{n1}*1a^{n1}*a=a^{n1}a^{n1+1}=a^{n1}a^{n}\).
Hope it's clear.
If we start at the "end" I can see how it is the same, however.. \(=a^{n1}a^{n+1}\) ? I don't see how these two share the ^(n 1)..



Math Expert
Joined: 02 Sep 2009
Posts: 44400

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
29 May 2013, 13:18



Intern
Joined: 20 Jun 2011
Posts: 45

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
29 May 2013, 13:25
Bunuel wrote: superpus07 wrote: Bunuel wrote: We are factoring our \(a^{n1}\) from \(a^{n1}a^n\) to get \(a^{n1}(1a)\):
\(a^{n1}(1a)=a^{n1}*1a^{n1}*a=a^{n1}a^{n1+1}=a^{n1}a^{n}\).
Hope it's clear.
If we start at the "end" I can see how it is the same, however.. \(=a^{n1}a^{n+1}\) ? I don't see how these two share the ^(n 1).. We have \(a^{n1}\) and \(a^{n}\). Now, \(a^{n}=a^{n1}*a\). P.S. We can factor out \(a^{n1}\) from \(a^{n1}a^{n+1}\) too: since \(a^{n+1}=a^{n1}a^2\), then \(a^{n1}a^{n+1}=a^{n1}(1a^2)\). For more on exponents check here: mathnumbertheory88376.htmlHope it helps. It's clear now. Thanks.



Current Student
Joined: 06 Sep 2013
Posts: 1948
Concentration: Finance

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
12 May 2014, 09:13
Good question +1. We are asked if n is odd given that 'a' and 'n' are integers. From statement 1 we have that 'n' must be odd so that if 'a' is negative then negative ^ even is positive > negative^ odd. Therefore sufficient. Statement 2 tells s that n must be odd otherwise if n is even then both numbers will have even exponent and there's no way that a^even> a^3even. Therefore D is the correct answer choice Hope it clarifies Cheers! J



Director
Joined: 10 Mar 2013
Posts: 579
Location: Germany
Concentration: Finance, Entrepreneurship
GPA: 3.88
WE: Information Technology (Consulting)

If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
18 Dec 2015, 08:33
conty911 wrote: If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n (2) a^n > a^(3n)
Please provide me pointers to any similar questions, also if possible
Can some one explain steps to tackle such problems with least amount of time.For eg. how to select the values for plugging etc. THanks In the MGMAT CATs there are lots of such caliber (time consuming ). Could not solve this one on the CAT because of the time issue, had to guess. (1) \(a^(n–1) > a^n\) here we have to check 2 cases n=ODD and n=EVEN ODD: In order this expression to be true a must be either \(ve or a fraction\), a is an integer, so a is ve EVEN: The only possibility this expression to hold true  a must be a positive fraction, BUT it's not a fraction, but an integer Thus, it holds true ONLY if n is ODD, Sufficient (2) Same game as above, and you get here that n must be an ODD integer Answer D
_________________
When you’re up, your friends know who you are. When you’re down, you know who your friends are.
Share some Kudos, if my posts help you. Thank you !
800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50 GMAT PREP 670 MGMAT CAT 630 KAPLAN CAT 660



Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5068
GPA: 3.82

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
19 Dec 2015, 05:28
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 a and n are integers and n > 1, is n odd? (1) a^(n–1) > a^n (2) a^n > a^(3n) In the original condition, there are 2 variables(a,n) and 1 equation(n>1, when it comes to inequality questions, an inequality also can be an equation.), which should match with the number equation. So you need 1 more equation. For 1) 1 equation, for 2 1equation, which is likely to make D the answer. In 1), from a^(n1)>a^n, if a>1, it is impossible because n1>n impossible as well as a=0,1. So, it has to be a<0. In this case, n=odd and n1=even. Therefore, a^(n1)>0, a^n<0 is yes, which is sufficient. In 2), from a^n>a^(3n), if a>1, it is impossible as well because n>3n is impossible so as a=0,1,1. So, it has to be a<1. In order to satisfy a^n>a^(3n) from 3n>n, only can n=odd be possible, which is yes and sufficient. Therefore, the answer is 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.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The oneandonly World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only $79 for 3 month Online Course" "Free Resources30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons  try it yourself"



Manager
Joined: 03 Jul 2015
Posts: 67
Concentration: Marketing, Finance

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
21 May 2016, 04:27
Can someone explain why my way of solving A is incorrect?
a^(n1)>a^n => a^n/a >a^n => a^n/a  a^n > 0 => a^n((1/a)a) > 0 => ((1a)/a) a^n > 0
Both the terms can either be positive or negative Taking the first case i.e. positive Say ((1a)/a) = 1 => a=1/2 Substituting the value of a in a^n For the second term i.e. a^n to be positive (1/2)^n can take the value of n as either positive or negative Why is then A sufficient?



Math Expert
Joined: 02 Sep 2009
Posts: 44400

Re: If a and n are integers and n > 1, is n odd? [#permalink]
Show Tags
21 May 2016, 05:06




Re: If a and n are integers and n > 1, is n odd?
[#permalink]
21 May 2016, 05:06



Go to page
1 2
Next
[ 26 posts ]



