GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Dec 2018, 01:16

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### Free GMAT Strategy Webinar

December 15, 2018

December 15, 2018

07:00 AM PST

09:00 AM PST

Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.

# If the greatest integer k for which 3^k is a factor of n! is 8, what

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Intern
Joined: 23 Mar 2014
Posts: 16
If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

Updated on: 18 Dec 2016, 21:51
4
11
00:00

Difficulty:

95% (hard)

Question Stats:

39% (02:13) correct 61% (02:17) wrong based on 219 sessions

### HideShow timer Statistics

If the greatest integer k for which 3^k is a factor of n! is 8, what is the largest possible value of p so that 5^p is a factor of n! ?

A) 2
B) 3
C) 4
D) 5
E) 6

Originally posted by Rocky1304 on 18 Dec 2016, 08:19.
Last edited by Bunuel on 18 Dec 2016, 21:51, edited 2 times in total.
Renamed the topic and edited the question.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8678
Location: Pune, India
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

03 Jan 2017, 02:46
4
2
Rocky1304 wrote:
If the greatest integer k for which 3^k is a factor of n! is 8, what is the largest possible value of p so that 5^p is a factor of n! ?

A) 2
B) 3
C) 4
D) 5
E) 6

We know that every multiple of 3 will give us a 3 in n!
We also know that every third multiple of 3 will give us another 3.

Let's assume a value of n and see how many 3s it has. Say, n = 15

Number of 3s in 15! = 15/3 + 5/3 = 6

But we need 8 3s. So we must jump to the next multiple of 3 i.e. 18. It gives us 2 3s so we get a total of 8 3s. The next multiple of 3 is 21 which will give another 3 but we have only 8 3s. So the maximum value of n can be 20.

The largest value of p so that 5^p is a factor of 20! is 20/5 = 4

For more on this concept, check:
https://www.veritasprep.com/blog/2011/0 ... actorials/
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

##### General Discussion
Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

18 Dec 2016, 08:53
2
Rocky1304 wrote:
If the greatest integer k for which $$3^k$$is a factor of n! is 8, what is the largest possible value of p
so that $$5^p$$ is a factor of n! ?

A) 2
B) 3
C) 4
D) 5
E) 6

$$\frac{n}{3} + \frac{n}{3^2} + \frac{n}{3^3} … = 8$$

$$n! = 18!, 19!, 20!$$

$$[\frac{18}{3}] + [\frac{18}{3^2}] = [\frac{19}{3}] + [\frac{19}{3^2}] = [\frac{20}{3}] + [\frac{20}{3^2}] = 8$$

From $$21!$$ we have additional factor of $$3$$ hence $$3^9$$.

$$18!$$ has $$5^3$$

$$19!$$ Still $$5^3$$

$$20!$$ One more factor of 5 – $$5^4$$

Max power of $$5$$ in $$n!$$ is $$4$$

Intern
Joined: 02 Jan 2017
Posts: 1
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

02 Jan 2017, 16:57
Hello everybody,

Is someone out there who can explain me the first step that was made? Why can you divide n by 3 and 3^2 to figure out what the maximum value of n might be? In general, I really don't understand the applied approach.

Kind regards

Posted from my mobile device
Manager
Joined: 17 May 2015
Posts: 249
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

02 Jan 2017, 21:55
Bonachhilfe wrote:
Hello everybody,

Is someone out there who can explain me the first step that was made? Why can you divide n by 3 and 3^2 to figure out what the maximum value of n might be? In general, I really don't understand the applied approach.

Kind regards

Posted from my mobile device

Hi Bonachhilfe,

Please refer the Factorial section on the following page:
http://gmatclub.com/forum/math-number-theory-88376.html

Thanks.
Intern
Joined: 08 Jul 2016
Posts: 18
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

26 Mar 2017, 05:13
I understand that n can be 18,19,20. However,the question never suggested that we have to consider the largest possible value of n. It says the largest possible value of k. So I substituted the value of n = 18 and not 20 while solving.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8678
Location: Pune, India
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

27 Mar 2017, 02:34
1
manishcmu wrote:
I understand that n can be 18,19,20. However,the question never suggested that we have to consider the largest possible value of n. It says the largest possible value of k. So I substituted the value of n = 18 and not 20 while solving.

This is what the question says: what is the largest possible value of p so that 5^p is a factor of n! ?

We need the largest possible value of p. We will get the largest possible value of p when n takes the largest possible value it can take. n can be 18, 19 or 20.
If n is 18, p is 3
If n is 19, p is 3.
If n is 20, then p is 4.
So 4 is the largest possible value of p.
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

Intern
Joined: 07 Dec 2016
Posts: 40
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

06 May 2017, 14:53
1
C

18!, 19! and 20! has 8 x 3's
So then 18! & 19! has 3 x 5's, while 20! has 4 x 5's
So 4 x 5's
_________________

Cheers!
If u like my post..... payback in Kudos!!

Senior Manager
Joined: 27 Dec 2016
Posts: 258
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

27 Jun 2017, 17:43
vitaliyGMAT wrote:
Rocky1304 wrote:
If the greatest integer k for which $$3^k$$is a factor of n! is 8, what is the largest possible value of p
so that $$5^p$$ is a factor of n! ?

A) 2
B) 3
C) 4
D) 5
E) 6

$$\frac{n}{3} + \frac{n}{3^2} + \frac{n}{3^3} … = 8$$

$$n! = 18!, 19!, 20!$$

$$[\frac{18}{3}] + [\frac{18}{3^2}] = [\frac{19}{3}] + [\frac{19}{3^2}] = [\frac{20}{3}] + [\frac{20}{3^2}] = 8$$

From $$21!$$ we have additional factor of $$3$$ hence $$3^9$$.

$$18!$$ has $$5^3$$

$$19!$$ Still $$5^3$$

$$20!$$ One more factor of 5 – $$5^4$$

Max power of $$5$$ in $$n!$$ is $$4$$

Hi, could you please explain how you got 18!, 19!, and 20! from your first step?

Thank You!
VP
Status: Learning
Joined: 20 Dec 2015
Posts: 1066
Location: India
Concentration: Operations, Marketing
GMAT 1: 670 Q48 V36
GRE 1: Q157 V157
GPA: 3.4
WE: Engineering (Manufacturing)
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

27 Jun 2017, 19:16
Imo C
We are given 3^k is a factor of n! for k=8
That means we have to find out which factorial contains that much power of 3
18! has 8 powers of 3.
But here is the catch 19! and 20! also have 8 powers of 3 because 19 and 20 are not factors of 3.
If we take 21! it contains additional 3 i.e 20!*21=20!*3*7
So we have to stop at 20!
Now we have to find out maximum power of 5 power of 5 in 18!=3
power of 5 in 19!=3
power of 5 in 20!=4
So we have 4 as the maximum power.
_________________

Please give kudos if you found my answers useful

Director
Joined: 14 Dec 2017
Posts: 518
Location: India
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what  [#permalink]

### Show Tags

16 Jul 2018, 10:01
Rocky1304 wrote:
If the greatest integer k for which 3^k is a factor of n! is 8, what is the largest possible value of p so that 5^p is a factor of n! ?

A) 2
B) 3
C) 4
D) 5
E) 6

Given: k = 8 is the greatest integer, such that n! is divisible by 3^k.

Hence we have 3*3*3....8 times in the expansion of n!

Hence we have n/3 + n/9 = 8, we get n = 18

However maximum value of n can be n = 20, as at n = 21, we will have 3 more than 8 times in n!

Now we need max power of 5 in 20!

Hence, 20/5 = 4

Thanks,
GyM
_________________
Re: If the greatest integer k for which 3^k is a factor of n! is 8, what &nbs [#permalink] 16 Jul 2018, 10:01
Display posts from previous: Sort by

# If the greatest integer k for which 3^k is a factor of n! is 8, what

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.