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# M17-24

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:01
2
10
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Difficulty:

75% (hard)

Question Stats:

45% (01:18) correct 55% (01:23) wrong based on 296 sessions

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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 $$n^3$$

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16 Sep 2014, 00:01
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Official Solution:

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

Statements (1) and (2) combined are insufficient. If $$N$$ is 6, both statements hold but the answer is "no". If $$N = 18$$, both statements hold and the answer is "yes".

Alternative Explanation

Algebraic approach:

Given: $$n=integer \gt 0$$. Question: is $$n=9k$$ (where $$k$$ is a positive integer)?

(1) 18 is a factor of $$n^2$$. Given, $$n^2=18p$$, where $$p$$ is a positive integer. So, $$n=3\sqrt{2p}$$. Now, since $$n$$ 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, ... Thus $$n$$ may or may not be a multiple of 9. Not sufficient.

(2) 27 is a factor of $$n^3$$. Given, $$n^3=27q$$, where $$q$$ is a positive integer. So, $$n=3\sqrt[3]{q}$$. Now, since $$n$$ 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, ... Thus $$n$$ 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. Hence, we just know that $$n$$ 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.

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06 Dec 2014, 16:09
1
I used the method described in books from MGMAT (prime box)
for N to be divisible by 9, it must contain in its prime box two of 3's

1) 18 is a factor of N^2
prime factorization of 18 = 2 * 3^2
n^2 = n * n
which means that one n, in its prime box, will take a two, and a three, since we do not know whether n might contain another 3, statement 1 is insufficient.
2) same method used, we have that n has only one 3, it might have more than one, but we do not know, hence the statement is insufficient.

1+2 -> we know for sure that n has one three in its prime box, but it might have more than one, therefore 1+2 insufficient.
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15 Apr 2017, 02:10
Please could you explain the concept of Prime boxes
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12 May 2017, 06:42
The answer must be option E.

From statement 1: n^2=18p ; n=3*sqrt(2p). Not sufficient as we can't be sure if the second part has a factor of 3 or not.
From statement 2: n^3=27q ; n=3*((q)^(1/3)). Not sufficient as we can't be sure if the second part has a factor of 3 or not.

Statement 1 & 2: n^5=(3^5)*(r) ---> n=3*((3)*(r^1/5)). Not sufficient as we can't be sure if the second part has a factor of 3 or not.

Option E is the correct answer
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10 Oct 2017, 07:03
1
Hello wwambugu14,

As a concept, the 'Prime Box' is very well explained in MGMAT strategy guide. It is explained in their 'Number properties' book. Give it a quick read and you would be able to handle problems such as this.

Also, there is a Thursdays with Ron video on this topic. You may want to have a look there as well. The link for it is -

wwambugu14 wrote:
Please could you explain the concept of Prime boxes

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31 Jul 2018, 05:21
I think this is a high-quality question and I agree with explanation.
Re M17-24 &nbs [#permalink] 31 Jul 2018, 05:21
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# M17-24

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