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If 14!/3^x is an integer, what is the maximum possible value of x?

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If 14!/3^x is an integer, what is the maximum possible value of x?  [#permalink]

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New post 06 Dec 2016, 07:17
00:00
A
B
C
D
E

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  15% (low)

Question Stats:

76% (00:40) correct 24% (00:49) wrong based on 101 sessions

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Re: If 14!/3^x is an integer, what is the maximum possible value of x?  [#permalink]

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New post 06 Dec 2016, 08:01
1
Bunuel wrote:
If 14!/3^x is an integer, what is the maximum possible value of x?

A. 2
B. 4
C. 5
D. 6
E. 9


MAximum possible value of 3 in 14! is -

\(\frac{14}{3} = 4\)
\(\frac{4}{3} = 1\)

So, The highest power of 3 in 14! will be 5 , hence answer will be (C) 5.


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Re: If 14!/3^x is an integer, what is the maximum possible value of x?  [#permalink]

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New post 06 Dec 2016, 08:30
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1
Bunuel wrote:
If 14!/3^x is an integer, what is the maximum possible value of x?

A. 2
B. 4
C. 5
D. 6
E. 9


The highest power of a prime \(p\) dividing \(n!\) is given by:
\(\Big[ \frac{n}{p}\Big]+\bigg[ \frac{n}{p^2}\bigg]+\bigg[ \frac{n}{p^3}\bigg]+...\) with \(\big[ x\big]\) is the largest integer \(y\) that \(y\leq x\)

Hence, the maximum possible value of \(x\) is
\(\Big[ \frac{14}{3}\Big]+\Big[ \frac{14}{3^2}\Big]=4+1=5\)

The answer is C
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Re: If 14!/3^x is an integer, what is the maximum possible value of x?  [#permalink]

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New post 24 Mar 2017, 07:59
nguyendinhtuong wrote:
Bunuel wrote:
If 14!/3^x is an integer, what is the maximum possible value of x?

A. 2
B. 4
C. 5
D. 6
E. 9


The highest power of a prime \(p\) dividing \(n!\) is given by:
\(\Big[ \frac{n}{p}\Big]+\bigg[ \frac{n}{p^2}\bigg]+\bigg[ \frac{n}{p^3}\bigg]+...\) with \(\big[ x\big]\) is the largest integer \(y\) that \(y\leq x\)

Hence, the maximum possible value of \(x\) is
\(\Big[ \frac{14}{3}\Big]+\Big[ \frac{14}{3^2}\Big]=4+1=5\)

The answer is C


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Re: If 14!/3^x is an integer, what is the maximum possible value of x?  [#permalink]

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New post 25 Mar 2017, 08:25
Count how many factors of 3 in 14!:
3 - 1
6 - 1
9 - 2
12 - 1

Answer is 5.
Re: If 14!/3^x is an integer, what is the maximum possible value of x? &nbs [#permalink] 25 Mar 2017, 08:25
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