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

It is currently 16 Oct 2019, 00:25

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

3^q is a factor of (300!/100!) where q is a positive integer. What is

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

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58369
3^q is a factor of (300!/100!) where q is a positive integer. What is  [#permalink]

Show Tags

New post 25 Jun 2019, 03:18
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

53% (02:09) correct 47% (01:46) wrong based on 60 sessions

HideShow timer Statistics

Director
Director
avatar
D
Joined: 20 Jul 2017
Posts: 890
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: 3^q is a factor of (300!/100!) where q is a positive integer. What is  [#permalink]

Show Tags

New post 25 Jun 2019, 05:19
1
1
Bunuel wrote:
3^q is a factor of (300!/100!) where q is a positive integer. What is the highest possible value of q?

(A) 67
(B) 89
(C) 100
(D) 114
(E) 148


Definitely a >2 min question

300!/100! = 101*102*103* . . . . . . . . *298*299*300

Multiples of 3 from 101 to 300 are
102, 105, 108 . . . . .300 (AP Series, Use last term to find number of multiples)
--> 300 = 102 + (x - 1)3
--> 198/3 = x - 1
--> x = 67

Multiples of 3^2 (9) from 101 to 300 are
108, 117, 126 . . . . . 297 (AP Series, Use last term to find number of multiples)
--> 297 = 108 + (y - 1)9
--> 189/9 = y - 1
--> y = 22

Multiples of 3^3 (27) from 101 to 300 are
108, 135, 162 . . . . . 297 (AP Series, Use last term to find number of multiples)
--> 297 = 108 + (z - 1)27
--> 189/27 = z - 1
--> z = 8

Multiples of 3^4 (81) from 101 to 300 are
162, 243 = 2

Multiples of 3^5 (243) from 101 to 300 are
243 = 1

Total factors = 67 + 22 + 8 + 2 + 1 = 100

So, 300!/100! = 3^100*k, where k is any non-multiple of 3

IMO Option C

Pls Hit Kudos if you like the solution
Director
Director
avatar
G
Joined: 22 Nov 2018
Posts: 557
Location: India
GMAT 1: 640 Q45 V35
GMAT 2: 660 Q48 V33
GMAT ToolKit User Premium Member
Re: 3^q is a factor of (300!/100!) where q is a positive integer. What is  [#permalink]

Show Tags

New post 25 Jun 2019, 07:31
3
1
Bunuel wrote:
3^q is a factor of (300!/100!) where q is a positive integer. What is the highest possible value of q?

(A) 67
(B) 89
(C) 100
(D) 114
(E) 148


No. of 3 in 100! is 100/3+100/9+100/27+100/81=33+11+3+1=48
No. of 3 in 300! is 300/3+300/9+300/27+300/81+300/243=100+33+11+3+1=148

So the 300!/100! prime factorized by 3 will be 3^148/3^48; Using exponent properties 3^100 IMO C
_________________
Give +1 kudos if this answer helps..!!
GMAT Club Legend
GMAT Club Legend
User avatar
D
Joined: 18 Aug 2017
Posts: 4997
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member
Re: 3^q is a factor of (300!/100!) where q is a positive integer. What is  [#permalink]

Show Tags

New post 26 Jun 2019, 03:13
1
2
Bunuel wrote:
3^q is a factor of (300!/100!) where q is a positive integer. What is the highest possible value of q?

(A) 67
(B) 89
(C) 100
(D) 114
(E) 148


factors of for 300! ; 300!/3 + 300!/9+300!/27+300!/81+300!/243 100+33+11+3+1=148

factors of 3 for 100! ; 100!/3+ 100!/9+... = 48
so 3^148/3^48 ; 3^100
IMO C
Target Test Prep Representative
User avatar
D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8069
Location: United States (CA)
Re: 3^q is a factor of (300!/100!) where q is a positive integer. What is  [#permalink]

Show Tags

New post 01 Jul 2019, 17:52
Bunuel wrote:
3^q is a factor of (300!/100!) where q is a positive integer. What is the highest possible value of q?

(A) 67
(B) 89
(C) 100
(D) 114
(E) 148


The number of factors of 3 in 300! is:

300/3 = 100

100/3 = 33

33/3 = 11

11/3 = 3

3/3 = 1

So there are 100 + 33 + 11 + 3 + 1 = 148 factors of 3 in 300!.

The number of factors of 3 in 100! is:

100/3 = 33

33/3 = 11

11/3 = 3

3/3 = 1

So there are 33 + 11 + 3 + 1 = 48 factors of 3in 300!.

Thus, there are 148 - 48 = 100 factors of 3 in (300!/100!), so the max value of q is 100.

Answer: C
_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
TTP - Target Test Prep Logo
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

GMAT Club Bot
Re: 3^q is a factor of (300!/100!) where q is a positive integer. What is   [#permalink] 01 Jul 2019, 17:52
Display posts from previous: Sort by

3^q is a factor of (300!/100!) where q is a positive integer. What is

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





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne