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What is the greatest positive integer n such that 3^n is a factor of

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Bunuel
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What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   18 Apr 2016, 00:57
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What is the greatest positive integer n such that 3^n is a factor of 36^100?

A. 100
B. 200
C. 300
D. 600
E. 900
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B
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chetan2u
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What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   18 Apr 2016, 01:01
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Bunuel wrote:
What is the greatest positive integer n such that 3^n is a factor of 36^100?

A. 100
B. 200
C. 300
D. 600
E. 900


Hi,
the Q basically asks us the number of 3 in \(36^{100}\)...
so lets factorize it--
\(36^{100} = (3^2*2^2)^{100} = 3^{2*100} * 2^{2*100} = 3^{200} * 2^{200}\)..
so n=200
B
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What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   18 Apr 2016, 12:21
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Bunuel wrote:
What is the greatest positive integer n such that 3^n is a factor of 36^100?

A. 100
B. 200
C. 300
D. 600
E. 900


\(36^{100}\) = \((3^{2}*2^{2})^{100}\) =\(3^{200} * 2^{200}\)

So, The Highest power of 3^n will be 3^200

Or, \(3^n\)= \(3^{200}\)

Hence n = 200 :-D :P :lol:
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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   18 Apr 2016, 12:51
Prime factorization out is the way to go here!

36 ^100

3^2 times 2^2 and both to the power of 100
[(3^2)*(2^2)]^100
giving you the beautifully simplistic answer of 3^200 and 2^200

Ans is 200
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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   18 Apr 2016, 18:00
What is the greatest positive integer n such that 3^n is a factor of 36^100?

A. 100
B. 200
C. 300
D. 600
E. 900

36 = 2^2 x 3^2.

36^100 will have 3^200. Hence the greatest integer value for n is 200.

Answer B.
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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   19 Apr 2016, 07:25
36^100 = {9*4)^100 = 3^200 (something)

So, The Highest power of 3^n will be 3^200
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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   20 Apr 2016, 14:27
Here using the property (ab)^n = a^n * b^n
we can say that 3^200 is a factored term => 200 i.e. B
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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   22 Jul 2019, 03:16
Bunuel wrote:
What is the greatest positive integer n such that 3^n is a factor of 36^100?

A. 100
B. 200
C. 300
D. 600
E. 900


36 = 3^2* 2^2
36^100 = 3^200* 2^200

IMO B

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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   30 Jul 2019, 09:45
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Bunuel wrote:
What is the greatest positive integer n such that 3^n is a factor of 36^100?

A. 100
B. 200
C. 300
D. 600
E. 900


36^100 = (2^2 x 3^2)^100 = 2^200 x 3^200, so the max value of n is 200.

Answer: B
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Re: What is the greatest positive integer n such that 3^n is a factor of [#permalink] New post   16 Apr 2024, 11:22
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