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Math Expert V
Joined: 02 Sep 2009
Posts: 58449
Is the positive integer n divisible by 6?  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 59% (01:59) correct 41% (02:00) wrong based on 207 sessions

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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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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)
Manager  Joined: 10 Sep 2014
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Re: Is the positive integer n divisible by 6?  [#permalink]

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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 Math Expert V
Joined: 02 Sep 2009
Posts: 58449
Re: Is the positive integer n divisible by 6?  [#permalink]

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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.

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

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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

Thanks
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Joined: 12 Aug 2015
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GRE 1: Q169 V154 Re: Is the positive integer n divisible by 6?  [#permalink]

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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
Insuff
Smash that A
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Re: Is the positive integer n divisible by 6?  [#permalink]

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Top Contributor
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

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

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_________________ Re: Is the positive integer n divisible by 6?   [#permalink] 07 Jun 2019, 04:02
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