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# If x is an integer, is x^3 divisible by 9?

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Intern
Joined: 17 Jun 2013
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If x is an integer, is x^3 divisible by 9? [#permalink]

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13 Jan 2014, 01:03
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If x is an integer, is x^3 divisible by 9?

(1) x^2 is divisible by 9.

(2) x^4 is divisible by 9.
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 33037
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Kudos [?]: 70598 [1] , given: 9854

If x is an integer, is x^3 divisible by 9? [#permalink]

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13 Jan 2014, 01:13
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If x is an integer, is x^3 divisible by 9?

Important thing to know when solving this problem is that exponentiation does not "produce" primes. For example, for integer x if x^(positive integer) is divisible by 3, then x must be divisible by 3. How else would 3 appear in x^(positive integer)?

(1) x^2 is divisible by 9 --> x^2 is divisible by 3^2 --> x is divisible by 3 --> x^3 is divisible by 3^3*k, thus it's divisible by 9 too. Sufficient.

(2) x^4 is divisible by 9 --> the same here: x^4 is divisible by 3^2 --> x is divisible by 3 --> x^3 is divisible by 3^3*k, thus it's divisible by 9 too. Sufficient.

Hope it's clear.
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Re: If x is an integer, is x^3 divisible by 9? [#permalink]

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01 Feb 2014, 07:47
Hi Bunuel, is it okay to reason stmt2 as below?

$$x^4$$ is divisible by 9.
then either
a. x is divisible by 9 or
b. $$x^3$$ is divisible by 9.

If a is true then b is also true.
if a is not true then b has to be true. therefore for $$x^4$$ to be divisible by 9, $$x^3$$ MUST be divisible by 9.
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Intern
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Re: If x is an integer, is x^3 divisible by 9? [#permalink]

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26 Sep 2014, 21:31
Bunuel wrote:
russ9 wrote:
If x is an integer, is x^3 divisible by 9?

1) x^2 is divisible by 9
2) x^4 is divisible by 9

Merging similar topics. Please refer to the discussion above.

Why cant we consider the case where x= sqrt 3; in that case x^3 will be divisible by 3sqrt 3
Math Expert
Joined: 02 Sep 2009
Posts: 33037
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Kudos [?]: 70598 [0], given: 9854

If x is an integer, is x^3 divisible by 9? [#permalink]

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27 Sep 2014, 01:23
Expert's post
ani781 wrote:
Bunuel wrote:
russ9 wrote:
If x is an integer, is x^3 divisible by 9?

1) x^2 is divisible by 9
2) x^4 is divisible by 9

Merging similar topics. Please refer to the discussion above.

Why cant we consider the case where x= sqrt 3; in that case x^3 will be divisible by 3sqrt 3

Because we are told that x is an integer, while $$\sqrt{3}$$ is not. Also, what does divisibility by $$3\sqrt{3}$$ has to do with the question? We need to find whether x^3 divisible by 9.

Also, note two things:
1. Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers (ALL GMAT divisibility questions are limited to positive integers only).

2. On the GMAT when we are told that $$a$$ is divisible by $$b$$ (or which is the same: "$$a$$ is multiple of $$b$$", or "$$b$$ is a factor of $$a$$"), we can say that:
(i) $$a$$ is an integer;
(ii) $$b$$ is an integer;
(iii) $$\frac{a}{b}=integer$$.

For more check [url=http://gmatclub.com/forum/divisibility-multiples-factors-tips-and-hints-174998.html]Divisibility/Multiples/Factors: Tips and hints
[/url].

Hope it helps.
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If x is an integer, is x^3 divisible by 9?   [#permalink] 27 Sep 2014, 01:23
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