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Math Expert V
Joined: 02 Sep 2009
Posts: 58991
Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 44% (00:47) correct 56% (00:43) wrong based on 79 sessions

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Competition Mode Question

Is X^N an integer?

(1) X is an integer
(2) N is an integer

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Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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Is X^N an integer?

(1) X is an integer
(4)^1/2 = 2 (integer), 4^2 = 16 (integer) but (4)^-2 (not integer).................INSUFFICIENT

(2) N is an integer
(1/2)^2 = i/4 (not integer) and 2^2= 4 = integer.............different answers possible......INSUFFICIENT

(1) + (2)
2^1 = integer but 2^-1 = not an integer. INSUFFICIENT

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Originally posted by EncounterGMAT on 09 Sep 2019, 22:01.
Last edited by EncounterGMAT on 10 Sep 2019, 23:19, edited 1 time in total.
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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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we are to determine if X^N is an integer.
From statement 1, we know that X is an integer. This is however insufficient since we don't know the value of N. If N is an integer, then X^N will be an integer, but if N is not an integer, say 0.5 and N is 3, then 3^0.5 is not an integer.

From statement 2, we know that N is an integer. Since no information is given in relation to X, there are infinite possibilities of X^N, some of which are integers with others not integers. Statement 2 is therefore not sufficient.

1+2 is sufficient because we know X and N are both integers, hence X^N will always be an integer.

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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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(1) X is an integer
N can take any value integer or non - integer.
a. N is an integer → $$X^N$$ is an integer. Example X = 2, N = 1, 2, 3 and so on. YES
b. N is non – integer → $$X^N$$ is non - integer. Example X = 2, N = 0.1, 0.2, 0.5 and so on. NO
INSUFFICIENT.

(2) N is an integer
X can take any value integer or non - integer.
a. X is an integer → $$X^N$$ is an integer. Example N = 2, X = 1, 2, 3 and so on. YES
b. X is non – integer → $$X^N$$ is non integer. Example N = 2, X = 0.1, 0.2, 0.5 and so on. NO
INSUFFICIENT.

Together 1) and 2)
Since both are integers Case ‘a’ of both statements suffices YES condition.

SUFFICIENT.
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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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1
#1
value of N not know insufficient as -ve integer value of N will be fraction
insufficient
#2
N is an integer ; value of X not know insufficeint
from 1&2
value of sign of x & n not know
insufficient
IMO E

Is X^N an integer?

(1) X is an integer
(2) N is an integer
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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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1
Quote:
Is $$X^N$$ an integer?

(1) X is an integer
(2) N is an integer

rule: "integer" = {positive, negative or zero}

(1) X is an integer: $$X,N=(2,1):X^N=2^1=2=integer…X,N=(2,-1):X^N=2^{(-1)}=0.5=not.integer$$
(2) N is an integer: $$X,N=(2,1):X^N=2^1=2=integer…X,N=(2,-1):X^N=2^{(-1)}=0.5=not.integer$$

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GMAT 1: 780 Q51 V45 GRE 1: Q170 V167 Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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1
Analyzing the question:
Note that x does not have to be an integer for this to be sufficient. For example, we may have x = sqrt(2) and N be any even number, and this would be sufficient. Also, N = 0 is automatically sufficient if x cannot be 0.

Statement 1:
Without knowing N this is insufficient. N can be 1/2, which is square of x. There's also the possibility that N is a negative integer.

Statement 2:
x can be sqrt(2) and N=1. Insufficient.

Combined:
Still insufficient as we can let N = -1 and x be any integer greater than 1 to create a fractional result. Answer is E.
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Originally posted by TestPrepUnlimited on 10 Sep 2019, 13:31.
Last edited by TestPrepUnlimited on 11 Sep 2019, 08:58, edited 1 time in total.
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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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Both statements are not sufficient as for X to the power of N to be an integer, we need X to be an integer and N to be a positive integer. From the given information, N can be either positive or negative and statement 1 is anyway not sufficient.

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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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Clearly statement 1,2 are not sufficient exclusively
If we consider both, for x=2, n=3 .....x^n is integer but if n=-2 then x^n is not integer
SO OA:E
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Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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Bunuel wrote:

Competition Mode Question

Is X^N an integer?

(1) X is an integer
(2) N is an integer

Is X^N an integer?

(1) X is an integer
$$2^2$$ is an integer
$$2^{-2}$$ is not an integer
NOT SUFFICIENT

(2) N is an integer
$$2^2$$ is an integer
$$2^{-2}$$ is not an integer
NOT SUFFICIENT

Combining (1) & (2)
$$2^2$$ is an integer
$$2^{-2}$$ is not an integer
NOT SUFFICIENT

IMO E
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Re: Is X^N an integer? (1) X is an integer (2) N is an integer  [#permalink]

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Is X^N an integer?

(1) X is an integer
N could be negative - which leads to a fraction.
(2) N is an integer
N could still be negative - which leads to a fraction. Re: Is X^N an integer? (1) X is an integer (2) N is an integer   [#permalink] 30 Sep 2019, 19:13
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