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• ### $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) # What is the smallest positive integer n for which 324 is a  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Intern Joined: 16 Jun 2010 Posts: 14 What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags Updated on: 01 Jun 2014, 08:38 1 15 00:00 Difficulty: 45% (medium) Question Stats: 62% (01:28) correct 38% (01:19) wrong based on 835 sessions ### HideShow timer Statistics What is the smallest positive integer n for which 324 is a factor of 6^n? A. 2 B. 3 C. 4 D. 5 E. 6 Originally posted by chintzzz on 19 Jun 2010, 00:09. Last edited by Bunuel on 01 Jun 2014, 08:38, edited 1 time in total. Edited the question. ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 50544 Re: please confirm answer [#permalink] ### Show Tags 19 Jun 2010, 00:30 3 4 chintzzz wrote: I tried solving the following question and arrived at (a). Please confirm if my answer is correct or the official answer (c) is correct. What is the smallest positive integer n for which 324 is a factor of 6 to the power of 2. A.2 B.3 C.4 D.5 E.6 Guess should be 6 to the power of $$n$$. Given: if $$6^n=324*k$$, then $$n_{min}=?$$ $$6^n=2^n*3^n=324*k=2^2*3^4*k$$ --> $$n_{min}=4$$ (for $$k=2^2=4$$). Answer: C. _________________ ##### Most Helpful Community Reply SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1827 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags 04 Mar 2014, 01:36 2 4 6^n / 324 324 = 6*6*3*3 We would require minimum n =4 so that 6*6*3*3 divides the number Answer = C _________________ Kindly press "+1 Kudos" to appreciate ##### General Discussion Manager Joined: 07 Oct 2006 Posts: 61 Location: India Re: please confirm answer [#permalink] ### Show Tags 19 Jun 2010, 00:49 Hi Bunuel, Even I thought there was some problem with the question. Good that you were able to find the mistake Thanks. _________________ ------------------------------------- Please give kudos, if my post is helpful. For English Grammar tips, consider visiting http://www.grammar-quizzes.com/index.html. Manager Joined: 18 Nov 2013 Posts: 66 Location: India GMAT Date: 12-26-2014 WE: Information Technology (Computer Software) Re: What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags 04 Mar 2014, 01:49 1 Clearly n = 2 and n = 3 would result in smaller numbers for 6^n/324. You can strike them off. Put n = 4, and divide 324 into further factors, we get 6*6*6*6/6*6*3*3. Hence, the answer is C. Intern Joined: 13 Dec 2013 Posts: 36 GMAT 1: 620 Q42 V33 Re: What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags 01 Jun 2014, 07:14 Hi, Bunuel, so the problem stem changed to "2" instead of "n"? thanks. Math Expert Joined: 02 Sep 2009 Posts: 50544 Re: What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags 01 Jun 2014, 08:40 Intern Joined: 10 Mar 2013 Posts: 11 Re: please confirm answer [#permalink] ### Show Tags 13 Jun 2014, 09:43 I understand the part where "6^n=2^n*3^n=324*k=2^2*3^4*k " however I don't understand how you get k = 4 in "--> n_{min}=4 (for k=2^2=4)." I understand the other solutions, however I would like to understand this one better, please. Thank you. Bunuel wrote: chintzzz wrote: I tried solving the following question and arrived at (a). Please confirm if my answer is correct or the official answer (c) is correct. What is the smallest positive integer n for which 324 is a factor of 6 to the power of 2. A.2 B.3 C.4 D.5 E.6 Guess should be 6 to the power of $$n$$. Given: if $$6^n=324*k$$, then $$n_{min}=?$$ $$6^n=2^n*3^n=324*k=2^2*3^4*k$$ --> $$n_{min}=4$$ (for $$k=2^2=4$$). Answer: C. Math Expert Joined: 02 Sep 2009 Posts: 50544 Re: please confirm answer [#permalink] ### Show Tags 13 Jun 2014, 09:59 2 1 farhanabad wrote: I understand the part where "6^n=2^n*3^n=324*k=2^2*3^4*k " however I don't understand how you get k = 4 in "--> n_{min}=4 (for k=2^2=4)." I understand the other solutions, however I would like to understand this one better, please. Thank you. Bunuel wrote: chintzzz wrote: I tried solving the following question and arrived at (a). Please confirm if my answer is correct or the official answer (c) is correct. What is the smallest positive integer n for which 324 is a factor of 6 to the power of 2. A.2 B.3 C.4 D.5 E.6 Guess should be 6 to the power of $$n$$. Given: if $$6^n=324*k$$, then $$n_{min}=?$$ $$6^n=2^n*3^n=324*k=2^2*3^4*k$$ --> $$n_{min}=4$$ (for $$k=2^2=4$$). Answer: C. $$6^n=2^n*3^n$$. $$324=2^2*3^4$$. Now, 6^n to be a multiple of 324, the powers of its primes must be at least as large as powers of primes in 324. Thus the least value of n for which 6^n is a multiple of 324 is 4. Does this make sense? _________________ Intern Joined: 10 Mar 2013 Posts: 11 Re: please confirm answer [#permalink] ### Show Tags 13 Jun 2014, 10:08 Yes, I guess I thought there was a magic formula where K = (2^n * 3^n)/((2^2*3^4). I wanted to learn how you would solve that. Thanks again. Manager Joined: 03 Jan 2015 Posts: 62 Concentration: Strategy, Marketing WE: Research (Pharmaceuticals and Biotech) Re: What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags 19 Jan 2015, 15:27 Bunuel wrote: chintzzz wrote: I tried solving the following question and arrived at (a). Please confirm if my answer is correct or the official answer (c) is correct. What is the smallest positive integer n for which 324 is a factor of 6 to the power of 2. A.2 B.3 C.4 D.5 E.6 Guess should be 6 to the power of $$n$$. Given: if $$6^n=324*k$$, then $$n_{min}=?$$ $$6^n=2^n*3^n=324*k=2^2*3^4*k$$ --> $$n_{min}=4$$ (for $$k=2^2=4$$). Answer: C. Hi, The question asks for minimum value of n^th power of 6. Because 324 = 2^2*3^2 i.e. 324 = 6^2*3^2. So shouldn't n = 2? TO EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12841 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 What is the smallest positive integer n for which 324 is a [#permalink] ### Show Tags 19 Jan 2015, 18:44 2 Hi All, This question can be solved with a bit of Number Property knowledge and TESTing THE ANSWERS. We're asked for the SMALLEST positive integer N for which 324 is a FACTOR of 6^N. In other words, which of these answers is SMALLEST and makes 6^N/324 is an integer. Since we're dealing with positives, and the answer choices are numbers, we can just "brute force" the answer choices until we find a match.... A: N = 2 6^2 = 36 36/324 is NOT an integer. Eliminate A. B: N = 3 6^3 = 216 216/324 is NOT an integer. Eliminate B. C: N = 4 6^4 = 1296 1296/324 might "look" tough, but consider the patterns.... "12" is divisibly by "3" (4 times) and "96" is divisibly by "24" (also 4 times), so 1296/324 = 4. This IS an integer, so C MUST be the answer. Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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05 Feb 2016, 09:23
The simplest way for me to solve the question is:

Prime factorisation of 324 = 2^2x3^4
Prime factorisation of 6 = 2x3

We need to have at least four factors of 2 and 3 on the number 6^N in order to end with an integer.

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Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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05 Feb 2016, 10:30
chintzzz wrote:
What is the smallest positive integer n for which 324 is a factor of 6^n?

A. 2
B. 3
C. 4
D. 5
E. 6

324 = 4*81 = 2^2*3^4

i.e. 6^n=2^n*3^n must have atleast 2^2*3^4

i.e. n Min must be 4

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Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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16 Sep 2018, 00:59
chintzzz wrote:
What is the smallest positive integer n for which 324 is a factor of 6^n?

A. 2
B. 3
C. 4
D. 5
E. 6

Dear Moderator,

Please retag this question from " Work /rate problems " to " Arithmetic ". Thank you.
_________________

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Posts: 50544
Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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16 Sep 2018, 01:01
stne wrote:
chintzzz wrote:
What is the smallest positive integer n for which 324 is a factor of 6^n?

A. 2
B. 3
C. 4
D. 5
E. 6

Dear Moderator,

Please retag this question from " Work /rate problems " to " Arithmetic ". Thank you.

________________________
Edited. The tags. Thank you.
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What is the smallest positive integer n for which 324 is a  [#permalink]

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16 Sep 2018, 01:21
1
chintzzz wrote:
What is the smallest positive integer n for which 324 is a factor of 6^n?

A. 2
B. 3
C. 4
D. 5
E. 6

factorize 324 ---> $$2^2*3^4$$

factorize 6 ---> $$2^1*3^1$$

$$\frac{2^1*3^1}{2^2*3^4}$$

now so that $$6^n$$ is divisible by 324 we need to add at least as many powers as 324 has

i.e. $$2^1*3^1$$ = $$2^2*3^4$$ (must be equal)

--> $$6^n$$ = $$2^1*2^1*3^1*3^3$$ = $$2^2*3^4$$ YAY
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Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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16 Sep 2018, 15:02
chintzzz wrote:
What is the smallest positive integer n for which 324 is a factor of 6^n?

A. 2
B. 3
C. 4
D. 5
E. 6

6^3=216=2/3*324
6^4=6*2/3*324=1296
1296/324=4
C
Re: What is the smallest positive integer n for which 324 is a &nbs [#permalink] 16 Sep 2018, 15:02
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