What is the smallest positive integer n for which 288 is a factor of

Question

What is the smallest positive integer  n  for which 288 is a factor of  6^n ?
 
(A) 2 
(B) 3 
(C) 4 
(D) 5 
(E) 6

BrentGMATPrepNow · Accepted Answer

-------ASIDE---------------------------------------
A lot of integer property questions can be solved using prime factorization. 
For questions involving factors, we can say:

If k is a factor of N, then k is "hiding" within the prime factorization of N

Consider these examples: 
 3  is a factor of 24, because 24 = (2)(2)(2) (3) , and we can clearly see the  3  hiding in the prime factorization. 
Likewise,  5  is a factor of 70 because 70 = (2) (5) (7)
And  8  is a factor of 112 because 112 = (2) (2)  (2)  (2) (7)
And  15  is a factor of 630 because 630 = (2)(3) (3)  (5) (7)
-----BACK TO THE QUESTION!---------------------

So, the original question can be rephrased to ask   "What's the smallest value of n so that 288 is hiding in the prime factorization of 6^n?   
Let's find out..

288 = (2)(2)(2)(2)(2)(3)(3)  
So, in order for 288 to be a factor of 6^n, the prime factorization of 6^n must contain  at least  five 2's and two 3's

So, for example, 6⁴ = (6)(6)(6)(6) = (2)(3)(2)(3)(2)(3)(2)(3). 
As you can see, 6⁴ contains at least two 3's, BUT it doesn't contain the five 2's we need. 
So, 288 is not a factor of 6⁴

On the other hand, 6⁵ = (6)(6)(6)(6)(6) = (2)(3)(2)(3)(2)(3)(2)(3)(2)(3) =   (2)(2)(2)(2)(2)(3)(3)  (3)(3)(3), which means  288  is definitely a factor of 6⁵

Answer: D

Archit3110 · Answer

BrentGMATPrepNow 
question is smallest +ve integer for which 288 is a factor of 6^n
6^n/2^3 *6^2

n has to be >2
least value of which would get us an integer value or say perfectly divide is 6^n/288 is when n = 5 and we get factor as 27 
option D

BrentGMATPrepNow · Answer

Good catch!!
I was reading the question the opposite way around. 
I read it as  6^n is a factor of 288  (not the other way around as intended)

I've fixed my solution accordingly.
Kudos for you!!!!!

PyjamaScientist · Answer

Hi  BrentGMATPrepNow , great solution, but I believe there's a small error in the translation of the problem at your end.

We need to the smallest value of ' n ' for which  288   will be a factor of  6^n . The answer would be  (D)   5  as  288 = 2^5 * 3^2 . And  6^n = (2*3)^n . For  288   to be a factor of  6^n = (2*3)^n , it should at least be equal to  5 .

Option (D)

BrentGMATPrepNow · Answer

Thanks for the heads up, but I think you may be referring to my original solution (which I have since edited following @Archit3110's earlier comments)

Thanks nonetheless. 
Kudos for you as well!!!

Ilanchezhiyan · Answer

288=2^5*3^2
For 6^n to be divided by 288
Soln:
6^n=2^n×3^n
The smallest value for n cannot be less than 5.

bumpbot · Answer

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What is the smallest positive integer n for which 288 is a factor of

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What is the smallest positive integer n for which 288 is a factor of

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