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# If a^n ≠ 0 and n is a positive integer, is n odd?

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Intern
Joined: 10 Sep 2013
Posts: 25
Location: India
Concentration: Accounting, Finance
GMAT Date: 10-25-2013
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If a^n ≠ 0 and n is a positive integer, is n odd?  [#permalink]

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Updated on: 20 Oct 2013, 03:43
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Difficulty:

65% (hard)

Question Stats:

67% (02:30) correct 33% (01:58) wrong based on 127 sessions

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If a^n ≠ 0 and n is a positive integer, is n odd?

(1) a^n + a^(n+1) < 0

(2) a is an integer.

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Originally posted by jellybean23 on 19 Oct 2013, 11:14.
Last edited by Bunuel on 20 Oct 2013, 03:43, edited 1 time in total.
Edited the question and added the OA.
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Re: If a^n ≠ 0 and n is a positive integer, is n odd?  [#permalink]

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19 Oct 2013, 12:42
1
1
Statement 1:

This essentially says $$a$$ is negative and thereby one of the terms $$a^n$$ or $$a^n^+^1$$ is negative (both can't be negative as either $$n$$ is even or $$n+1$$ is even)

Let's assign $$a=-2$$. So $$a^n + a^n^+^1 < 0$$ only when $$n$$ is even. But what happens if $$a$$ is not an integer? Assign $$a=-0.1$$, then for the equation to be less than zero, $$n$$ has to be odd. INSUFFICIENT

Statement 2:

This tells us that $$a$$ is an integer, but does not say anything about $$n$$. INSUFFICIENT

Statement 1 & 2: as discussed under Statement 1, when $$a$$ is a negative integer, $$n$$ has to be even for the equation to be less than zero. So we know $$n$$ is not odd. SUFFICIENT

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Re: If a^n ≠ 0 and n is a positive integer, is n odd?  [#permalink]

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19 Oct 2013, 19:11
2
n >0
Statement 1 :
a^n + a^(n+1) < 0
lets put n =1 (odd) , a+a^2 <0 .................. , that means 0 > a > -1,
lets put n=2 (even) , a^2 + a^3 < 0 ........... , that means a < 0
but no info about a so no conclusion about n.. Insufficient..

Statement 2 :
a is an integer , but no conclusion about n... insufficient...

Taking both statements together....
only n=2 valid .... which gives n is not an odd no..
Intern
Joined: 10 Sep 2013
Posts: 25
Location: India
Concentration: Accounting, Finance
GMAT Date: 10-25-2013
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Re: If a^n ≠ 0 and n is a positive integer, is n odd?  [#permalink]

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19 Oct 2013, 20:50
+1 Kudos !!!!
great explanation by both of you.
Thanks
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Posts: 52285
Re: If a^n ≠ 0 and n is a positive integer, is n odd?  [#permalink]

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20 Oct 2013, 03:54
If a^n ≠ 0 and n is a positive integer, is n odd?

a^n ≠ 0 and n is a positive integer implies that a ≠ 0.

(1) a^n + a^(n+1) < 0 --> $$(a+1)a^n<0$$. Two cases:

$$a+1>0$$ and $$a^n<0$$. From first inequality we have that $$a>-1$$. If a is a negative number (say -1/2), then $$(negative)^n$$ to be negative n must be odd.

$$a+1<0$$ and $$a^n>0$$. From first inequality we have that $$a<-1$$, so a is a negative number. Now, $$(negative)^n$$ to be positive n must be even.

Not sufficient.

(2) a is an integer. Clearly insufficient.

(1)+(2) Consider the same two cases but now take into account that a is an integer:

$$a+1>0$$ and $$a^n<0$$. From first inequality we have that $$a>-1$$. Since we know that $$a\neq{0}$$, then a must be a positive integer (1, 2, 3, ...). Next, $$(positive)^n$$ cannot be negative, thus this case is out.

$$a+1<0$$ and $$a^n>0$$. From first inequality we have that $$a<-1$$, so a is a negative integer. Now, $$(negative)^n$$ to be positive n must be even.

Sufficient.

Hope it's clear.
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Re: If a^n ≠ 0 and n is a positive integer, is n odd?  [#permalink]

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30 Mar 2018, 22:37
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).

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Re: If a^n ≠ 0 and n is a positive integer, is n odd? &nbs [#permalink] 30 Mar 2018, 22:37
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# If a^n ≠ 0 and n is a positive integer, is n odd?

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