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# If N is a positive integer, is 9 a factor of N?

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If N is a positive integer, is 9 a factor of N? [#permalink]

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24 May 2010, 22:58
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If N is a positive integer, is 9 a factor of N?

(1) 18 is a factor of N^2
(2) 27 is a factor of N^3
[Reveal] Spoiler: OA

Last edited by Bunuel on 01 Sep 2017, 09:30, edited 2 times in total.
Renamed the topic, edited the question and added the OA.

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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25 May 2010, 03:43
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Jinglander wrote:
can someone show me how to answer this without having to find actual number

If N is a positive integer, is 9 a factor of N?

1. 18 is a factor of N^2
2. 27 is a factor of N^3

Algebraic approach:

Given $$n=integer>0$$. Q: is $$n=9k$$ (where $$k$$ is a positive integer)?

(1) 18 is a factor of n^2 --> $$n^2=18p$$, where $$p$$ is a positive integer --> $$n=3\sqrt{2p}$$ --> as $$n$$ is an integer $$\sqrt{2p}$$ must be an integer too, taking the even integer values --> so basically $$n=3\sqrt{2p}$$ would be a multiple of 3*2=6, taking the following values: 6, 12, 18, ... So $$n$$ may or may not be a multiple of 9. Not sufficient.

(2) 27 is a factor of n^3 --> $$n^3=27q$$, where $$q$$ is a positive integer --> $$n=3\sqrt[3]{q}$$ --> as $$n$$ is an integer $$\sqrt[3]{q}$$ must be an integer too --> so basically $$n=3\sqrt[3]{q}$$ would be just a multiple of 3: 3, 6, 9, 12, ... So $$n$$ may or may not be a multiple of 9. Not sufficient.

(1)+(2) From (1) $$n$$ is a positive multiple of 6 and from (2) $$n$$ is a positive multiple of 3 --> $$n$$ is a positive multiple of 6, so $$n$$ could be 6, 12, 18, ... Thus $$n$$ may or may not be a multiple of 9. Not sufficient.

Also:
 ! Please post PS questions in the PS subforum: gmat-problem-solving-ps-140/Please post DS questions in the DS subforum: gmat-data-sufficiency-ds-141/No posting of PS/DS questions is allowed in the main Math forum.

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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25 May 2010, 07:11
I will post in the right place from now on. Also in part 1 why only take the even integers to realize is was a multiple of 6

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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25 May 2010, 07:20
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Jinglander wrote:
I will post in the right place from now on. Also in part 1 why only take the even integers to realize is was a multiple of 6

Posted from my mobile device

$$n=3\sqrt{2p}=integer$$ --> $$\sqrt{2p}=integer$$, which basically means that 2p is a even perfect square: 4, 16, 36, 64 --> so $$\sqrt{2p}$$ can be $$\sqrt{4}=2$$, $$\sqrt{16}=4$$, $$\sqrt{36}=6$$, $$\sqrt{64}=8$$, ... even numbers --> $$n=3*even$$, so n is multiple of 6 for sure.

Hope it's clear.
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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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25 May 2010, 08:27
Thanks

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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27 May 2010, 10:10
I feel stupid for asking, but why are prime numbers only positive? Can't negative numbers be prime numbers, i.e. -2 or -3? Is this something I just have to know and not worry too much about?

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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27 May 2010, 11:13
hcb wrote:
I feel stupid for asking, but why are prime numbers only positive? Can't negative numbers be prime numbers, i.e. -2 or -3? Is this something I just have to know and not worry too much about?

By definition a prime number is a positive integer that has no positive integer divisors other than 1 and itself.

So just remember it and don't worry about it.

(The main reason we have this restrictions in definition is that if we remove them there will be some problems with fundamental theorem of arithmetic (unique prime factorization theorem), so removing them would require many other adjustment).
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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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27 May 2010, 11:15
OK - thanks Bunuel

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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15 Dec 2015, 08:55
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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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20 Aug 2016, 09:11
what on earth are N2 and N3

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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20 Aug 2016, 10:33
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stonecold wrote:
what on earth are N2 and N3

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Lol.. Try to understand the emotions

N2 and N3 means N^2 and N^3.
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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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20 Aug 2016, 11:49
abhimahna wrote:
stonecold wrote:
what on earth are N2 and N3

Regards
Stonecold

Lol.. Try to understand the emotions

N2 and N3 means N^2 and N^3.

Then E is the answer i guess
Logic used => N and N^p has same prime factor.
A better way => Take N =6 and Smash that E
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30 Aug 2017, 23:03
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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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01 Sep 2017, 09:26
I really don't understand why it is writen as N2 instead of N^2

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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01 Sep 2017, 09:49
Condition of N is not clear so E will be ans

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Re: If N is a positive integer, is 9 a factor of N? [#permalink]

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01 Sep 2017, 09:51
Jinglander wrote:
If N is a positive integer, is 9 a factor of N?

(1) 18 is a factor of N^2
(2) 27 is a factor of N^3

(1) 18 is a factor of N^2

Let N = 6.............. 18 factor of (6)^2................Answer is No

Let N =18.............. 18 factor of (18)^2..............Answer is Yes

Insufficient

(2) 27 is a factor of N^3

Let N = 6.............. 27 factor of (6)^3................Answer is No

Let N =18.............. 27 factor of (18)^3..............Answer is Yes

Insufficient

Combine 1 & 2

Use same examples above.....No clear answer

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Re: If N is a positive integer, is 9 a factor of N?   [#permalink] 01 Sep 2017, 09:51
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