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# If p is an integer and 3^p is a factor of 300!, what is the maximum

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Senior PS Moderator
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If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 11:33
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53% (01:25) correct 47% (02:00) wrong based on 103 sessions

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If p is an integer and 3^p is a factor of 300!, what is the maximum possible value of p?

A.100
B.101
C.133
D.147
E.148

Source: Experts Global

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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 11:49
1
1
Number of primes p in n! -> Remainder addition of [n/p + n/p^2 + n/p^3...]

Number of threes in 300! ->
=>300/3 + 300/9 + 300/27 + 300/81 + 300/243
=> 100 + 33 + 11 + 3 + 1
=> 148

Smash that E.
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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 11:59
stonecold wrote:
Number of primes p in n! -> Remainder addition of [n/p + n/p^2 + n/p^3...]

Number of threes in 300! ->
=>300/3 + 300/9 + 300/27 + 300/81 + 300/243
=> 100 + 33 + 11 + 3 + 1
=> 148

Smash that E.

Hi stonecold i think you smashed correctly

can you please advice (if you know any) some article about how to solve such kind of questions "3^p is a factor of 300!"

i had no idea how to approach it,

thanks!
Intern
Joined: 20 Nov 2017
Posts: 16
Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 16:32
Can this method be used as 2^n aswell?

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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 20:42
1
Hey ra5867 & dave13

Please find below the link to the thread on factorials
https://gmatclub.com/forum/everything-a ... 85592.html

Under this, you will find the detailed explanation under
Finding the powers of a prime number p, in the n!

Hope this helps!
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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 22:29
pushpitkc wrote:
If p is an integer and 3^p is a factor of 300!, what is the maximum possible value of p?

A.100
B.101
C.133
D.147
E.148

Source: Experts Global

$$\frac{300}{3} = 100$$
$$\frac{100}{3} = 33$$
$$\frac{33}{3} = 11$$
$$\frac{11}{3} = 3$$
$$\frac{3}{3} = 1$$

So, the maximum possible value of p will be 100 + 33 + 11 + 3 + 1 = 148 , answer must be (E)

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Joined: 02 Sep 2009
Posts: 64068
Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 22:31
ra5867 wrote:
Can this method be used as 2^n aswell?

Sent from my SM-N910C using GMAT Club Forum mobile app

Everything about Factorials on the GMAT
Power of a Number in a Factorial Problems
Trailing Zeros Problems

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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21 Jan 2018, 23:02
dave13 wrote:
stonecold wrote:
Number of primes p in n! -> Remainder addition of [n/p + n/p^2 + n/p^3...]

Number of threes in 300! ->
=>300/3 + 300/9 + 300/27 + 300/81 + 300/243
=> 100 + 33 + 11 + 3 + 1
=> 148

Smash that E.

Hi stonecold i think you smashed correctly

can you please advice (if you know any) some article about how to solve such kind of questions "3^p is a factor of 300!"

i had no idea how to approach it,

thanks!

Check this post for a detailed discussion:
https://www.veritasprep.com/blog/2011/0 ... actorials/
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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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28 Jan 2018, 10:20
The official answer is E!

Official Explanation by Experts Global

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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum  [#permalink]

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06 Aug 2019, 04:04
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Re: If p is an integer and 3^p is a factor of 300!, what is the maximum   [#permalink] 06 Aug 2019, 04:04

# If p is an integer and 3^p is a factor of 300!, what is the maximum

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