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

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Joined: 02 Sep 2009
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If 14!/3^x is an integer, what is the maximum possible value of x?  [#permalink]

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06 Dec 2016, 07:17
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15% (low)

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

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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

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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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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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06 Dec 2016, 08:30
1
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
$$\Big[ \frac{14}{3}\Big]+\Big[ \frac{14}{3^2}\Big]=4+1=5$$

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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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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
$$\Big[ \frac{14}{3}\Big]+\Big[ \frac{14}{3^2}\Big]=4+1=5$$

Hi Thanks for explanation
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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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25 Mar 2017, 08:25
Count how many factors of 3 in 14!:
3 - 1
6 - 1
9 - 2
12 - 1

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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