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# What is the smallest positive integer n for which 324 is a

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Joined: 16 Jun 2010
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What is the smallest positive integer n for which 324 is a  [#permalink]

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Updated on: 01 Jun 2014, 09:38
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Difficulty:

45% (medium)

Question Stats:

63% (01:13) correct 37% (00:58) wrong based on 707 sessions

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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, 01:09.
Last edited by Bunuel on 01 Jun 2014, 09:38, edited 1 time in total.
Edited the question.
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19 Jun 2010, 01:30
3
3
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 $$k=2^2=4$$).

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

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04 Mar 2014, 02:36
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

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19 Jun 2010, 01:49
Hi Bunuel,

Even I thought there was some problem with the question. Good that you were able to find the mistake

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

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04 Mar 2014, 02:49
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.
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Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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01 Jun 2014, 08:14
Hi,

Bunuel, so the problem stem changed to "2" instead of "n"?

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

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01 Jun 2014, 09:40
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13 Jun 2014, 10:43
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 $$k=2^2=4$$).

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13 Jun 2014, 10:59
2
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 $$k=2^2=4$$).

$$6^n=2^n*3^n$$.

$$324=2^2*3^4$$.

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?
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13 Jun 2014, 11:08
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.
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Re: What is the smallest positive integer n for which 324 is a  [#permalink]

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19 Jan 2015, 16:27
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 $$k=2^2=4$$).

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?

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

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19 Jan 2015, 19:44
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.

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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, 10:23
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, 11:30
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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