Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 01 May 2016, 05:02

### 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 June
Open Detailed Calendar

# If N is the product of all positive integers less than 31,

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 18 Sep 2009
Posts: 359
Followers: 3

Kudos [?]: 302 [6] , given: 2

If N is the product of all positive integers less than 31, [#permalink]

### Show Tags

19 Oct 2010, 22:57
6
KUDOS
13
This post was
BOOKMARKED
00:00

Difficulty:

75% (hard)

Question Stats:

50% (02:20) correct 50% (01:27) wrong based on 422 sessions

### HideShow timer Statictics

If N is the product of all positive integers less than 31, than what is the greatest integer k for which N/18^k is an integer?

A. 3
B. 6
C. 7
D. 14
E. 26
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 32549
Followers: 5634

Kudos [?]: 68351 [7] , given: 9797

### Show Tags

20 Oct 2010, 03:47
7
KUDOS
Expert's post
13
This post was
BOOKMARKED
If N is the product of all positive integers less than 31, than what is the greatest integer k for which N/18^k is an integer?

A. 3
B. 6
C. 7
D. 14
E. 26

Given: $$n=30!$$. Question: if $$\frac{30!}{18^k}=integer$$ then $$k_{max}=?$$

We should determine the highest power of 18 in 30!.

$$18=2*3^2$$, so we should find the highest powers of 2 and 3 in 30!:

Highest power of 2 in 30!: $$\frac{30}{2}+\frac{30}{4}+\frac{30}{8}+\frac{30}{16}=15+7+3+1=26$$, --> $$2^{26}$$;

Highest power of 3 in 30!: $$\frac{30}{3}+\frac{30}{9}+\frac{30}{27}=10+3+1=14$$ --> $$3^{14}$$;

$$n=30!=2^{26}*3^{14}*p$$, where $$p$$ is the product of other multiples of 30! (other than 2 and 3) --> $$n=30!=(2*3^{2})^7*2^{19}*p=18^7*2^{19}*p$$ --> so the highest power of 18 in 30! is 7 --> $$\frac{30!}{18^k}=\frac{18^7*2^{19}*p}{18^k}=integer$$ --> $$k=7$$.

Hope it's clear.
_________________
Current Student
Joined: 06 Sep 2013
Posts: 2035
Concentration: Finance
GMAT 1: 770 Q0 V
Followers: 42

Kudos [?]: 455 [0], given: 355

Re: If N is the product of all positive integers less than 31, [#permalink]

### Show Tags

08 Oct 2013, 13:05
1
This post was
BOOKMARKED
TomB wrote:
If N is the product of all positive integers less than 31, than what is the greatest integer k for which N/18^k is an integer?

A. 3
B. 6
C. 7
D. 14
E. 26

Hi all, this was my approach for solving this.

Basically the question is asking us what is the highest power of 18 that can give us a number that is a factor of 31!

So remember 18 after prime factorization is (3^2)(2). Now there are going to be less factors of 3 in 31!, than factors of 2.
Therefore, lets find how many factors of 3 in 31! We can use this quick method

31/3^1 = 10
31/3^2= 3
31/3^3=1
Sum = 14
Just ignore the remainders.

So we have that 3^14 must be the least. Now don't forget that 18 is 3^2k so k must be ONLY 7, because 2k will give us the 14.

Hope it helps

Bunuel could you please validate this one? Thank you
Cheers
J

Last edited by jlgdr on 12 Feb 2014, 07:40, edited 1 time in total.
Manager
Joined: 07 May 2013
Posts: 109
Followers: 0

Kudos [?]: 17 [1] , given: 1

Re: If N is the product of all positive integers less than 31, [#permalink]

### Show Tags

12 Oct 2013, 19:33
1
KUDOS
According to Bunuel $$18=2*3^2$$

I listed 14 3's
14 3's- 3 3 3 3 3 3 3 3 3 3 3 3 3 3

26 2's- 2 2 2 2 2 2 2 2 2 2 2 2 2 2 and so on

notice that 7 such combinations of 18 are possible.
so answer= $$18^7$$
Senior Manager
Joined: 08 Apr 2012
Posts: 464
Followers: 1

Kudos [?]: 36 [0], given: 58

### Show Tags

13 Nov 2013, 13:43
Bunuel wrote:
If N is the product of all positive integers less than 31, than what is the greatest integer k for which N/18^k is an integer?

A. 3
B. 6
C. 7
D. 14
E. 26

Given: $$n=30!$$. Question: if $$\frac{30!}{18^k}=integer$$ then $$k_{max}=?$$

We should determine the highest power of 18 in 30!.

$$18=2*3^2$$, so we should find the highest powers of 2 and 3 in 30!:

Highest power of 2 in 30!: $$\frac{30}{2}+\frac{30}{4}+\frac{30}{8}+\frac{30}{16}=15+7+3+1=26$$, --> $$2^{26}$$;

Highest power of 3 in 30!: $$\frac{30}{3}+\frac{30}{9}+\frac{30}{27}=10+3+1=14$$ --> $$3^{14}$$;

$$n=30!=2^{26}*3^{14}*p$$, where $$p$$ is the product of other multiples of 30! (other than 2 and 3) --> $$n=30!=(2*3^{2})^7*2^{19}*p=18^7*2^{19}*p$$ --> so the highest power of 18 in 30! is 7 --> $$\frac{30!}{18^k}=\frac{18^7*2^{19}*p}{18^k}=integer$$ --> $$k=7$$.

Hope it's clear.

Hi Bunuel,
The logic here is the same as the logic in finding terminating "0" of a number right?
But instead of checking for the level of "5" we do it for "3^2", right?
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 9245
Followers: 454

Kudos [?]: 115 [0], given: 0

Re: If N is the product of all positive integers less than 31, [#permalink]

### Show Tags

01 Jan 2015, 07:29
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 9245
Followers: 454

Kudos [?]: 115 [0], given: 0

Re: If N is the product of all positive integers less than 31, [#permalink]

### Show Tags

15 Jan 2016, 01:55
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If N is the product of all positive integers less than 31,   [#permalink] 15 Jan 2016, 01:55
Similar topics Replies Last post
Similar
Topics:
6 What is the product of all positive odd integers less than 1 3 05 Oct 2013, 02:36
69 If n is a positive integer and the product of all integers 44 28 Dec 2009, 11:47
If n is a positive integer and the product of all integers 4 07 Nov 2009, 11:02
n is a positive integer, and k is the product of all integer 9 28 Sep 2009, 19:49
If n is a positive integer and the product of all the intege 7 17 Oct 2007, 08:44
Display posts from previous: Sort by

# If N is the product of all positive integers less than 31,

 Powered by phpBB © phpBB Group and phpBB SEO 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®.