Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 61385

If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 00:22
Question Stats:
33% (01:43) correct 67% (01:41) wrong based on 58 sessions
HideShow timer Statistics
Competition Mode Question If a and b are nonzero integers, is \(a^b\) an integer? (1) \(b^a\) is negative (2) \(a^b\) is negative Are You Up For the Challenge: 700 Level Questions
Official Answer and Stats are available only to registered users. Register/ Login.
_________________



CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 3180
Location: India
GMAT: INSIGHT
WE: Education (Education)

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 00:45
Quote: If a and b are nonzero integers, is \(a^b\) an integer?
(1) \(b^a\) is negative (2) \(a^b\) is negative Question: is \(a^b\) an integer?Statement 1: \(b^a\) is negativei.e. b is Negative and a is ODD integer Case 1: \(3^{1} = 1/3\) i.e. NOT an integer Case 2: \(1^{1} = 1\) i.e. an integer NOT SUFFICIENT Statement 2: \(a^b\) is negativei.e. a is Negative and b is ODD integer Case 1: \((3)^{1} = 1/3\) i.e. NOT an integer Case 2: \((1)^{1} = 1\) i.e. an integer NOT SUFFICIENT Combining the two statementsi.e. a and b both are Negative and both are ODD integer Case 1: \((3)^{1} = 1/3\) i.e. NOT an integer Case 2: \((1)^{1} = 1\) i.e. an integer NOT SUFFICIENT Answer: Option E
_________________
Prosper!!!GMATinsightBhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhihttp://www.GMATinsight.com/testimonials.htmlClick here for Our VERBAL & QUANT private tutoring package detailsACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Senior Manager
Joined: 27 Feb 2014
Posts: 339
Location: India
Concentration: General Management, International Business
GMAT 1: 570 Q49 V20
GPA: 3.97
WE: Engineering (Education)

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 01:46
When a and b are nonzero integers, consider following two cases for a^b: Case 1: when b is positive integer, then a^b is integer Case 2: when b is negative integer, then a^b is a fraction ie noninteger
To determine whether a^b is an integer or not, we only require sign of b without sign of a. Hence, rephrasing the question as what is the sign of integer b?
(1) When b^a is negative only when b is negative integer. Sufficient
(2) When a^b is negative, a is negative integer, whereas b can be positive or negative integer. Insufficient
A is correct



Senior Manager
Joined: 20 Mar 2018
Posts: 465
Location: Ghana
Concentration: Finance, Real Estate

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 02:45
If a and b are nonzero integers, is a^b an integer? Constraint: a and b ain’t zero Asked : a^b = Integer ,Yes /No?
(1) b^a is negative b^a < 0 ,then b must be negative and a must be odd .: when a = 1 ,b = 1 a^b = (1)^(1) = 1 = Integer? Yep! when a= 3 ,b =2 a^b= (3)^(2) = 1/(3^2)= 0.111=Integer? Nope! (Not sufficient)
(2) a^b is negative a^b < 0, then a must be negative and b must be odd. But a^b can be negative Integer such as 1 or negative nonInteger such as 1/2 (Not sufficient)
(1+2) We know a= odd and negative, b= odd and negative Still not sufficient to determine wether a^b =Integer/NonInteger Eg 1^(1) vs 3^(1) (Not sufficient)
Hit that E
Posted from my mobile device



VP
Joined: 24 Nov 2016
Posts: 1224
Location: United States

If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
Updated on: 22 Jan 2020, 03:07
Quote: If a and b are nonzero integers, is a^b an integer?
(1) b^a is negative (2) a^b is negative (1) b^a is negative suficb^a<0 then b<0, and a=anything≠0. a,b=1,1: 1^1=1 and 1^1=integer a,b=3,2: 2^3=1/8 and 3^2=1/9≠integer (2) a^b is negative insufica^b<0 then a<0, and b=anything≠0. a,b=3,2: 3^2=9=integer a,b=3,2: 3^2=1/9≠integer (1&2) insufica<0 and b<0; a,b=1,1: 1^1=1=integer a,b=2,3: 2^3=1/8≠integer Ans (E)
Originally posted by exc4libur on 21 Jan 2020, 05:24.
Last edited by exc4libur on 22 Jan 2020, 03:07, edited 1 time in total.



Manager
Joined: 30 Jul 2019
Posts: 101
Location: Viet Nam
Concentration: Technology, Entrepreneurship
GPA: 2.79
WE: Education (NonProfit and Government)

If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
Updated on: 21 Jan 2020, 23:35
If a and b are nonzero integers, is \(a^b\) an integer?
(1) \(b^a\) is negative (2) \(a^b\) is negative
(1) \(b^a\) is negative => b<0; a is odd if a = 1, b = 1 => \(a^b = 1\) is an integer if a = 3, b = 3 => \(a^b = 1/9 \) is not an integer => Not Suff (2) \(a^b\) is negative => a<0; b is odd if a = 1, b = 1 = > \(a^b = 8\) is an integer if a = 3, b = 3 => \(a^b = 1/9 \) is not an integer => Not suff Combine (1) and (2) => a<0, b<0, a and b are odd Same example as only (1) or only (2) => Not suff
=> Choice E
Originally posted by ostrick5465 on 21 Jan 2020, 10:12.
Last edited by ostrick5465 on 21 Jan 2020, 23:35, edited 1 time in total.



Director
Joined: 07 Mar 2019
Posts: 705
Location: India
WE: Sales (Energy and Utilities)

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 11:20
If a and b are nonzero integers, is \(a^b\) an integer? (1) \(b^a\) is negative \(b^a\) < 0 only when b < 0 where a > 0 OR a < 0 (a is odd integer) Example: a = 1 b = 2 \(b^a = (2)^{1} = \frac{1}{2}\); \(a^b = (1)^{2} = 1\) YES a = 3, b = 10 \(b^a = (10)^{3} = \frac{1}{1000}\); \(a^b = (3)^{10} = \frac{1}{3^{10}}\) NO INSUFFICIENT. (2) \(a^b\) is negative \(a^b\) < 0 only when a < 0 where b > 0 OR b < 0 (b is odd integer) Example: a = 1 b = 3 \(a^b = (1)^{3} = 1\) YES a = 3, b = 1 \(a^b = (3)^{1} = \frac{1}{3}\) NO INSUFFICIENT. Together 1 and 2. Both a < 0 and b < 0 and are odd. So, Example: a = 1 b = 3 \(b^a = (3)^{1} = \frac{1}{3}\); \(a^b = (1)^{3} = 1\) YES a = 3, b = 1 \(b^a = (1)^{3} = 1\); \(a^b = (3)^{1} = \frac{1}{3}\) NO a = 3, b = 5 \(b^a = (5)^{3} = \frac{1}{125}\); \(a^b = (3)^{5} = \frac{1}{3^5}\) NO INSUFFICIENT. Answer E.
_________________
Ephemeral Epiphany..!
GMATPREP1 590(Q48,V23) March 6, 2019 GMATPREP2 610(Q44,V29) June 10, 2019 GMATPREPSoft1 680(Q48,V35) June 26, 2019



CR Forum Moderator
Joined: 18 May 2019
Posts: 707

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 12:33
Given that a and b are nonzero integers, we are to determine if a^b is an integer.
Statement 1: b^a is negative. For b^a to be negative, then a must be odd and b must be a negative number. Let a=3 and b=2 then b^a = 2^3 = 8. a^b = 3^2 = 1/9 No. However when a=1, and b=3, then b^a = 3^1 = 1/3 a^b = 1^3 = 1. Yes. Statement 1 is insufficient.
Statement 2: a^b is negative. Statement 1 implies a is negative and b is odd. when a =1 and b=3, a^b = 1 yes. However when a=2 and b=3, then a^b = 2^3 = 1/8, No. Statement 2 is insufficient.
1+2 To satisfy both 1 and 2, we must have both a and b as odd negative values. a=1 and b=3, a^b = 1^3 = 1 and b^a = 3^1 = 1/3. The answer is yes. when a=b=3, then a^b = b^a = 1/27. The answer is No.
Both statements even taken together are not sufficient.
The answer is E.



Director
Joined: 25 Jul 2018
Posts: 560

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 12:57
If a and b are nonzero integers, is \(a^{b} \) an integer?
(Statement1): \(b^{a}\) is negative. > b <0, a could be any odd integer. if b=1 and a = 1, then \(a^{b}= 1^{1}= 1\) (yes) if b=2 and a=3, then \(a^{b}= 3^{2}= \frac{1}{9}\) (NO) Insufficient
(Statement2): \(a^{b}\) is negative. > a <0, b could be any odd integer. if a=1 and b= 1, then \(a^{b}= (1)^{1}= 1\) (Yes) if a= 2 and b=3, then \(a^{b}= (2)^{3}= \frac{1}{8}\) (NO) Insufficient
Taken together 1&2, a,b any odd negative integers (a<0, b<0) if a=1 and b=1, then \(a^{b}= (1)^{1}= 1 \)(Yes) if a=3 and b=3, then \(a^{b}= (3)^{3}= \frac{1}{27}\) (No) Insufficient
The answer is E.



Director
Joined: 30 Sep 2017
Posts: 668
GMAT 1: 720 Q49 V40
GPA: 3.8

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 14:56
a and b are nonzero integers, is a^b an integer?
(1) b^a is negative If a=1 and b=1, then b^a is negative and a^b is an integer If a=3 and b=3, then b^a is negative but a^b is not an integer NOT SUFFICIENT (2) a^b is negative If a=1 and b=1, then a^b is negative and a^b is an integer If a=3 and b=3, then a^b is negative but a^b is not an integer NOT SUFFICIENT
1)+2) Using the same scenarios, it is still uncertain whether a^b is an integer. NOT SUFFICIENT
FINAL ANSWER IS (E)



Manager
Joined: 26 Dec 2017
Posts: 54

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 18:46
Ans: E
a)b^a= ve (1)^(1)=1(b=a=1), so, a^b=1 int yes (1)^(3)=1 (b=1,a=3), so, a^b=1/3 not int no not sufficient
b)a^b=ve (1)^(1)=1(a=b=1) yes (3)^(3)=1/27(a=b=3) No not sufficient
taking both a & b (1)^(1)=1(a=b=1) yes (3)^(3)=1/27(a=b=3) No
Not possible



Manager
Status: Student
Joined: 14 Jul 2019
Posts: 171
Location: United States
Concentration: Accounting, Finance
GPA: 3.9
WE: Education (Accounting)

Re: If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 21:35
If a and b are nonzero integers, is a^b an integer?
(1) \(b^a\) is negative (2)\( a^b\) is negative
For \(a^b\) to be an integer, b has to be positive. but if we can get to know whether a =1, the sign of b will not be a matter.
1) from this statement, we can deduce that b is negative and a is an odd integer. when a = 1, then \( a^b \) can be an integer, but for any other value of a, \(a^b\) is not an integer. insufficient.
2) a is negative and b is odd. no information about the sign of b. insufficient.
Together, a and b are both negative and odd. For any value of a except 1, \(a^b\) won't be an integer. but we don't know about the value of a. insufficient.
E is the answer.



GMAT Club Legend
Joined: 18 Aug 2017
Posts: 5888
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)

If a and b are nonzero integers, is a^b an integer?
[#permalink]
Show Tags
21 Jan 2020, 22:39
possible when a=1 and b = 3 and a = and b =3
#1b^a is negative yes /no insufficient #2 a^b is negative yes/no insufficient nothing in common IMO E
If a and b are nonzero integers, is a^b an integer?
(1) b^a is negative (2) a^b is negative




If a and b are nonzero integers, is a^b an integer?
[#permalink]
21 Jan 2020, 22:39






