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

 It is currently 22 Oct 2019, 04:54 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ

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

Hide Tags

Senior Manager  P
Status: Manager
Joined: 02 Nov 2018
Posts: 280
The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ  [#permalink]

Show Tags

1
5 00:00

Difficulty:   65% (hard)

Question Stats: 30% (01:52) correct 70% (02:12) wrong based on 20 sessions

HideShow timer Statistics

The number 5^867 is between 2^2013 and 2^2014. How many pairs of integers (m,n) are there such that$$1 ≤ m ≤ 2012$$ and $$5^n < 2^m$$ < 2^ m+2 < 5^n+1?

(A) 278
(B) 279
(C) 280
(D) 281
(E) 282

_________________
Give a kudos if u find my post helpful. kudos motivates active discussions GMAT Club Legend  D
Joined: 18 Aug 2017
Posts: 5031
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ  [#permalink]

Show Tags

The number 5^867 is between 2^2013 and 2^2014. How many pairs of integers (m,n) are there such that$$1 ≤ m ≤ 2012$$ and $$5^n < 2^m$$ < 2^ m+2 < 5^n+1?

(A) 278
(B) 279
(C) 280
(D) 281
(E) 282

chetan2u ; please advise on solution to this question.. Math Expert V
Joined: 02 Aug 2009
Posts: 8006
The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ  [#permalink]

Show Tags

1
Archit3110 wrote:
The number 5^867 is between 2^2013 and 2^2014. How many pairs of integers (m,n) are there such that$$1 ≤ m ≤ 2012$$ and $$5^n < 2^m$$ < 2^ m+2 < 5^n+1?

(A) 278
(B) 279
(C) 280
(D) 281
(E) 282

chetan2u ; please advise on solution to this question.. Hi Archit,

Let us first understand the question..
$$5^n < 2^m< 2^ {m+2} < 5^{n+1}$$ .. So we are looking for consecutive power of 5( n to n+1) that contains 3 consecutive powers of 2 ( m to m+2)
If you want to do it, it can have two solutions..
(I) Calculation intensive....
Try to get a pattern ..
$$5^0,2^1,2^2,5^1,2^3,2^4,5^2,2^5,2^6,5^3,2^7,2^8,2^9,5^3,2^{10},2^{11}, 5^4$$
I have not gone beyond this, but you likely to see a pattern after a certain time

(II) A more elegant way
The series $$5^0,2^1,2^2,5^1,2^3,2^4,5^2,2^5,2^6,5^3,2^7,2^8,2^9,5^3,2^{10},2^{11}, 5^4$$ shows that
we can have 2 powers of 2( as between $$5^1$$ and $$5^2$$, we have $$2^3, 2^4$$) or
3 powers of 2 between consecutive powers of 5( as between $$5^3$$ and $$5^4$$, we have $$2^7,2^8,2^9$$).
Let there be x gaps that has 2 powers of 2 and y gaps that have 3 powers of 2.
SO when we add these GAPS, they should be equal to the total power of 5, which is 867.=>$$x+y=867$$
But in x gaps there are two powers of 2, that is 2x powers of 2, and y gaps have three power of 2, that is 3y, so when we add them we should get 2013 => $$2x+3y=2013$$
Multiply $$x+y=867$$by 2 and subtract from $$2x+3y=2013..=> 2x+3y-2(x+y)=2013-2*867....y=2013-1734=279$$

B
_________________
GMAT Club Legend  D
Joined: 18 Aug 2017
Posts: 5031
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ  [#permalink]

Show Tags

chetan2u wrote:
Archit3110 wrote:
The number 5^867 is between 2^2013 and 2^2014. How many pairs of integers (m,n) are there such that$$1 ≤ m ≤ 2012$$ and $$5^n < 2^m$$ < 2^ m+2 < 5^n+1?

(A) 278
(B) 279
(C) 280
(D) 281
(E) 282

chetan2u ; please advise on solution to this question.. Hi Archit,

Let us first understand the question..
$$5^n < 2^m< 2^ {m+2} < 5^{n+1}$$ .. So we are looking for consecutive power of 5( n to n+1) that contains 3 consecutive powers of 2 ( m to m+2)
If you want to do it, it can have two solutions..
(I) Calculation intensive....
Try to get a pattern ..
$$5^0,2^1,2^2,5^1,2^3,2^4,5^2,2^5,2^6,5^3,2^7,2^8,2^9,5^3,2^{10},2^{11}, 5^4$$
I have not gone beyond this, but you likely to see a pattern after a certain time

(II) A more elegant way
The series $$5^0,2^1,2^2,5^1,2^3,2^4,5^2,2^5,2^6,5^3,2^7,2^8,2^9,5^3,2^{10},2^{11}, 5^4$$ shows that
we can have 2 powers of 2( as between $$5^1$$ and $$5^2$$, we have $$2^3, 2^4$$) or
3 powers of 2 between consecutive powers of 5( as between $$5^3$$ and $$5^4$$, we have $$2^7,2^8,2^9$$).
Let there be x gaps that has 2 powers of 2 and y gaps that have 3 powers of 2.
SO when we add these GAPS, they should be equal to the total power of 5, which is 867.=>$$x+y=867$$
But in x gaps there are two powers of 2, that is 2x powers of 2, and y gaps have three power of 2, that is 3y, so when we add them we should get 2013 => $$2x+3y=2013$$
Multiply $$x+y=867$$by 2 and subtract from $$2x+3y=2013..=> 2x+3y-2(x+y)=2013-2*867....y=2013-1734=279$$

B

chetan2u ; appreciate and thanks for the solution..
honestly its a question which has really gone beyond my scope of understanding of gmat quant course... ... do such questions really come in actual exam is there really a boundary or limit to which gmat quant questions can be restricted to also how did you arrive on the highlighted part .. gaps would be 867?
Math Expert V
Joined: 02 Aug 2009
Posts: 8006
Re: The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ  [#permalink]

Show Tags

1
[quote="Archit3110"]

You are NOT likely to see such questions in GMAT..
now we are looking at $$5^0...5^1...5^2...5^3........5^{866}...5^{867}$$, so 867 gaps
Now $$(5^0-2^1,2^2-5^1)...(5^1-2^3,2^4-5^2)...(5^2-2^5,2^6-5^3)....$$ are the x gaps that have 2 power of 2
$$5^3-2^7,2^8,2^9-5^4.......$$ are the y gaps that have 3 power of 2
_________________
GMAT Club Legend  D
Joined: 18 Aug 2017
Posts: 5031
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ  [#permalink]

Show Tags

chetan2u wrote:
Archit3110 wrote:

You are NOT likely to see such questions in GMAT..
now we are looking at $$5^0...5^1...5^2...5^3........5^{866}...5^{867}$$, so 867 gaps
Now $$(5^0-2^1,2^2-5^1)...(5^1-2^3,2^4-5^2)...(5^2-2^5,2^6-5^3)....$$ are the x gaps that have 2 power of 2
$$5^3-2^7,2^8,2^9-5^4.......$$ are the y gaps that have 3 power of 2

really glad to see the bold part ok understood ; thanks a a lot .. Re: The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ   [#permalink] 26 Mar 2019, 10:23
Display posts from previous: Sort by

The number 5^867 is between 2^2013 and 2^2014. How many pairs of integ

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

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