Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 20 Jul 2019, 17:14

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

For positive integers n and m, is m!<3^n ?

  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: 56304
For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 26 Sep 2016, 03:33
3
8
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

53% (02:12) correct 47% (01:42) wrong based on 151 sessions

HideShow timer Statistics


Retired Moderator
avatar
G
Joined: 26 Nov 2012
Posts: 586
Re: For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 26 Sep 2016, 05:46
1
Bunuel wrote:
For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3


Stat 1: n = m

m! < 3^n ...this statement will work till 5 = n = m

120 < 3*3*3*3*3

Now if m = 7 and n =7 .

5040 > 2187....so insufficient.

Stat 2: we don't have relationship between n and m ...insufficient.

Both : if we take n > 3...i.e. 4 to 6...it is fine but if it is n = 7...Insufficient.

IMO option E.
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7764
For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 25 Feb 2018, 08:43
2
Bunuel wrote:
For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3



Nothing much from \(m! < 3^n\)..

let's see statements

1) \(n=m\)..
\(n=m=1.. 1!<3^1\) so \(1< 3\) ..yes
each term thereafter the left term m! will increase 4 times then 5 times and even 1000times when it is 1000! WHEREAS right will increase ONLY 3 times with each increase
So at some place Left side will become GREATER than Right side
BOTH \(m! < 3^n\) and \(m! > 3^n\) possible
Insuff

2) \(n>3..\)
Nothing about m
insufficient

combined..
lets check for \(m=n=4\)..
\(4!<3^4...24<81...\)yes
But at higher values we know that \(m!>3^n\)
insuff

E
_________________
Retired Moderator
avatar
D
Joined: 25 Feb 2013
Posts: 1196
Location: India
GPA: 3.82
GMAT ToolKit User Reviews Badge
For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 26 Feb 2018, 09:35
Bunuel wrote:
For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3


Is \(m!<3^n=>\frac{m!}{3^n}<1\) -------(1)

now \(m!=m(m-1)(m-2)(m-3)....3*2*1\) and \(3^n=3*3*3\)..... to n terms

so \(\frac{m!}{3^n}=\frac{m(m-1)(m-2)(m-3)....3*2*1}{3*3*3*3.....n terms}\)

Notice that in the above fraction only numerators 2 & 1 are less than 3 and rest all numerators will be either greater than 3 or equal to 3. So if in the numerator we have a number, such as 9, that can take care of the two 3's in the denominator then equation (1) will be definitely greater than 1

Statement 1: \(n=m=4\), then equation

then \(\frac{m!}{3^n}=\frac{4*3*2*1}{3*3*3*3}\), clearly \(\frac{4}{3}\) is marginally greater than \(1\), \(\frac{3}{3}=1\) but \(\frac{2}{3}\) & \(\frac{1}{3}\) are less than \(1\). Hence this fraction will be definitely less than \(1\). So we have a Yes for equation (1). but if \(n=m=10\)

then \(\frac{m!}{3^n}=\frac{10*9*8*7*6*5*4*3*2*1}{3*3*3*3*3*3*3*3*3*3}\), clearly \(10\) & \(9\) will take care of 3's corresponding to numerators \(2\) & \(1\). Hence this number will definitely be greater than \(1\). so we have a No for equation (1). Insufficient

Statement 2: nothing mentioned about \(m\). Insufficient

Combining (1) & (2): As we can see from statement 1, we will have both the scenarios, hence Insufficient

Option E
Intern
Intern
User avatar
Joined: 07 Dec 2017
Posts: 3
Re: For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 26 Feb 2018, 11:39
Can anyone help me where everyone seeing options???
I can't see them with the question.
_________________
--
Akanksha Sharma
Retired Moderator
avatar
D
Joined: 25 Feb 2013
Posts: 1196
Location: India
GPA: 3.82
GMAT ToolKit User Reviews Badge
Re: For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 26 Feb 2018, 11:41
SharmaAkanksha wrote:
Can anyone help me where everyone seeing options???
I can't see them with the question.


Hi,

In the DS question options are standard. Hence they are not represented again

Posted from my mobile device
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56304
Re: For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 26 Feb 2018, 11:45
SharmaAkanksha wrote:
Can anyone help me where everyone seeing options???
I can't see them with the question.


This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following posts:
ALL YOU NEED FOR QUANT.
Ultimate GMAT Quantitative Megathread

Hope this helps.
_________________
Manager
Manager
avatar
S
Joined: 10 Apr 2018
Posts: 130
Concentration: Leadership, Operations
GMAT 1: 600 Q44 V28
GPA: 3.56
WE: Engineering (Computer Software)
Reviews Badge
Re: For positive integers n and m, is m!<3^n ?  [#permalink]

Show Tags

New post 22 Aug 2018, 04:10
SharmaAkanksha wrote:
Can anyone help me where everyone seeing options???
I can't see them with the question.


The most common strategy is answers will be either from AD or BCE.
These are DS questions. And are about half of the quant questions (15 out of 31) will be of this type.
Happy to help.
_________________


The Graceful
----------------------------------------------------------
Every EXPERT was a beginner once...
Don't look at the clock. Do what it does, keep going
..
To achieve great things, two things are needed:a plan and not quite enough time - Leonard Bernstein.
GMAT Club Bot
Re: For positive integers n and m, is m!<3^n ?   [#permalink] 22 Aug 2018, 04:10
Display posts from previous: Sort by

For positive integers n and m, is m!<3^n ?

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





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