If a and b are nonzero integers, is a^b an integer?

Question

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

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GMATinsight · Answer

Question : is  a^b  an integer?

Statement 1 :  b^a  is negative

i.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 negative

i.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 statements

i.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

QuantMadeEasy · Answer

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 non-integer

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

Staphyk · Answer

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 non-Integer 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/Non-Integer 
Eg -1^(-1) vs -3^(-1)
(Not sufficient)

Hit that  E

Posted from my mobile device

exc4libur · Answer

(1) b^a is negative   sufic

b^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   insufic

a^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)   insufic

a<0 and b<0;
a,b=-1,-1: -1^-1=-1=integer
a,b=-2,-3: -2^-3=-1/8≠integer

Ans (E)

ostrick5465 · Answer

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 non-Integer 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/Non-Integer
Eg -1^(-1) vs -3^(-1)
(Not sufficient)

Hit that E

Posted from my mobile device

unraveled · Answer

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 non-Integer 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/Non-Integer
Eg -1^(-1) vs -3^(-1)
(Not sufficient)

Hit that E

Posted from my mobile device

eakabuah · Answer

Given that a and b are non-zero 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.

lacktutor · Answer

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.

freedom128 · Answer

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)

debjit1990 · Answer

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

minustark · Answer

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.

Archit3110 · Answer

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

Thekingmaker · Answer

This isn't very tough nut to crack if you commute all the different combination that validate let me show you:-
A b^a negative could be -1^1 accoedingly 1^-1 integer however {-1/2)^3 accordingly 3^-1/2 not an integer insuff 
B similarly we get it also as insuff even combined doesn't make the matter any better 
Therefore IMO E

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If a and b are nonzero integers, is a^b an integer?

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If a and b are nonzero integers, is a^b an integer?

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