Is the positive integer n divisible by 6?

Question

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 .

BrentGMATPrepNow · Accepted Answer

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

kinghyts · Answer

Statement 1: n^2/180 is an integer.

n^2 = 180*a = 6^2*5 *a

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)

DangerPenguin · Answer

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

Bunuel · Answer

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 	imes 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 	imes 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 	imes 2 	imes 2 	imes 2 	imes 3 	imes 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.

kinghyts · Answer

Hello DangerPenguin,

If n^2/180 is an integer, you can say n^2 = 180 * a

The minimum value could be 180 * 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

stonecold · Answer

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

Is the positive integer n divisible by 6?

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