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Is the positive integer n divisible by 6?

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Is the positive integer n divisible by 6?  [#permalink]

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New post 12 Nov 2014, 09:05
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Tough and Tricky questions: Divisibility.



Is the positive integer \(n\) divisible by \(6\)?


(1) \(\frac{n^2}{180}\) is an integer.

(2) \(\frac{144}{n^2}\) is an integer.

Kudos for a correct solution.

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Re: Is the positive integer n divisible by 6?  [#permalink]

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New post 12 Nov 2014, 20:27
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Statement 1: n^2/180 is an integer.

n^2 = 180*a = 6^2*5 *a [Where a is an integer]

Therefore n has to be a multiple of 6. Sufficient

Statement 2 : 144/n^2 is an integer.

n = 1, when n is not divisible by 6
n= 12, when n is divisible by 6
Therefore, insufficient

Therefore the answer has to be A)
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Re: Is the positive integer n divisible by 6?  [#permalink]

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New post 13 Nov 2014, 07:24
kinghyts wrote:
Statement 1: n^2/180 is an integer.

n^2 = 180*a = 6^2*5 *a [Where a is an integer]

Therefore n has to be a multiple of 6. Sufficient

Statement 2 : 144/n^2 is an integer.

n = 1, when n is not divisible by 6
n= 12, when n is divisible by 6
Therefore, insufficient

Therefore the answer has to be A)


Hey King,

I was having trouble with this question and didn't understand where you got "a" from. Also, how did you know "n=1 when n is not divisible by 6" ?

Thanks :)
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Re: Is the positive integer n divisible by 6?  [#permalink]

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New post 13 Nov 2014, 08:09
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Bunuel wrote:

Tough and Tricky questions: Divisibility.



Is the positive integer \(n\) divisible by \(6\)?


(1) \(\frac{n^2}{180}\) is an integer.

(2) \(\frac{144}{n^2}\) is an integer.

Kudos for a correct solution.


Official Solution:

We must determine whether \(n\) is evenly divisible by \(6\). In other words, we must determine whether \(n\) has a factor of 2 and a factor of 3 (since \(2 \times 3 = 6\)). If \(n\) is evenly divisible by 6, then \(\frac{n}{6} = k\) for some integer \(k\), and we can rewrite \(n\) as \(6k\).

Statement 1 says that \(\frac{n^2}{180}\) is an integer. In order for \(n\) to be divisible by 6, \(n^2\) must have two factors of 6, since \((6k)^2 = 36k^2\). Since \(n^2\) is divisible by 180, 180 is a factor of \(n^2\), and all factors of 180 are also factors of \(n^2\). Because 180 has 36 as a factor, and \(36 = 6 \times 6\), \(n^2\) has two factors of 6. Thus, \(n\) has 6 as a factor, and \(n\) is divisible by 6. Statement 1 is sufficient to answer the question. Eliminate answer choices B, C, and E. The correct answer choice is either A or D.

Statement 2 says that \(\frac{144}{n^2}\) is an integer. The prime factorization of 144 is \(2 \times 2 \times 2 \times 2 \times 3 \times 3\). This means that \(n^2\) can contain up to four factors of 2 and two factors of 3 (note that any given factor must occur an even number of times in the prime factorization of \(n^2\)). If \(n = 4\), the statement is satisfied, since \(\frac{144}{n^2} = \frac{144}{4^2} = \frac{144}{16} = 9\). In this case, \(n\) is NOT divisible by 6. However, \(n\) could also equal 6, since \(\frac{144}{n^2} = \frac{144}{6^2} = \frac{144}{36} = 4\). In this case, \(n\) is divisible by 6. Statement 2 does NOT provide sufficient information to answer the question.


Answer: A.
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Re: Is the positive integer n divisible by 6?  [#permalink]

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New post 13 Nov 2014, 10:21
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Hello DangerPenguin,

If n^2/180 is an integer, you can say n^2 = 180 * a [Where is a can be an integer e.g: 1,2,3 ... ]

The minimum value could be 180 * 1 [ where a=1 ] = 36 * 5 = 6^2 * 5

But 180 , which is 6^2 * 5 is not a perfect square. Therefore, the minimum value of n^2 should be = 180 * 5 = 6^2*5^2

Therefore n = 6 * 5 . Now this number n is divisible by 6. Isn't it ?


Coming to your next question, n = 1 and n =12 are the two values I picked to ensure that we are not getting definitive "Yes" or "No" by using the statement 2.

Let me know if you still have doubts

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Re: Is the positive integer n divisible by 6?  [#permalink]

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New post 24 Aug 2016, 05:46
To check => n is divisible by 6 => it must be divisible y 2 and 3
Statement 1 => Using the property that Both n and n^x have the same prime factors.=> n must have 2,3,5 as its prime factors
Although it may have more prime factors too but these are a must .
Hence n is clearly divisible by 6 => suff
Statement 2 => n=1 => NO n is not divisible by 6
N=6 =< YES n is divisible by 6
Contradiction
Insuff
Smash that A
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Re: Is the positive integer n divisible by 6?  [#permalink]

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New post 20 Apr 2018, 06:54
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Bunuel wrote:

Tough and Tricky questions: Divisibility.



Is the positive integer \(n\) divisible by \(6\)?


(1) \(\frac{n^2}{180}\) is an integer.

(2) \(\frac{144}{n^2}\) is an integer.

Kudos for a correct solution.


A lot of integer property questions can be solved using prime factorization.

For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)

----------------------------------------------------------

Okay, onto the question!

Target question: Is the positive integer n divisible by 6?

Statement 1: n²/180 is an integer
This tells us that n² is DIVISIBLE by 180
This means that 180 is "hiding in the prime factorization of n²
180 = (2)(2)(3)(3)(5)
So, n² = (2)(2)(3)(3)(5)(?)(?)(?)(?)

Aside: the (?)'s represent other possible primes in the prime factorization of n²

Rewrite as (n)(n) = [(2)(3)(5)(?)(?)][(2)(3)(5)(?)(?)]
This tells us that we can be certain that n = (2)(3)(5)(?)(?)
At this point it is clear that n is divisible by 6
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 144/n² is an integer
There are several values of n that satisfy this condition. Here are two:
Case a: n = 1. Notice that 144/1² = 144, and 144 is an integer. In this case n is NOT divisible by 6
Case b: n = 6. Notice that 144/6² = 4, and 4 is an integer. In this case n IS divisible by 6
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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Re: Is the positive integer n divisible by 6? &nbs [#permalink] 20 Apr 2018, 06:54
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