Author 
Message 
TAGS:

Hide Tags

Director
Joined: 18 Apr 2005
Posts: 531
Location: Canuckland

How many digits 2^100 has? [#permalink]
Show Tags
12 Jun 2005, 14:41
2
This post received KUDOS
31
This post was BOOKMARKED
Question Stats:
45% (01:23) correct 55% (01:27) wrong based on 538 sessions
HideShow timer Statistics
How many digits 2^100 has? A) 31 B) 35 C) 50 D) 99 E) 101
Official Answer and Stats are available only to registered users. Register/ Login.



Director
Joined: 01 Feb 2003
Posts: 824
Location: Hyderabad

Re: PS V, a toughy [#permalink]
Show Tags
12 Jun 2005, 15:30
sparky wrote: How many digits 2^100 has?
A) 31 B) 35 C) 50 D) 99 E) 101
hmm....am sure there is a better way to do this, but 
2^10 = 1.024 * 10^3 => 2^100 = (1.024)^10 * 10^30
therefore 31 digits would be my best guess



Director
Joined: 18 Apr 2005
Posts: 531
Location: Canuckland

Re: PS V, a toughy [#permalink]
Show Tags
12 Jun 2005, 15:38
1
This post was BOOKMARKED
Vithal wrote: sparky wrote: How many digits 2^100 has?
A) 31 B) 35 C) 50 D) 99 E) 101 hmm....am sure there is a better way to do this, but  2^10 = 1.024 * 10^3 => 2^100 = (1.024)^10 * 10^30 therefore 31 digits would be my best guess
cool way to do it, i didn't think about this one
I did it like this
at max 1 digit can be added after every third power, eg 2 4 8, 16 32 64, 128 etc
so max possible digit based on this observation can be (100 1)/3 + 1 = 34
so the answer 31 comes naturally.



Senior Manager
Joined: 30 May 2005
Posts: 370

1
This post was BOOKMARKED
I solved it like sparky did  2^100 = (2^3)^33 * 2
Number of digits is less than 34. So A is the only choice by elimination.



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1837
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: How many digits 2^100 has? A) 31 B) 35 C) 50 D) 99 E) [#permalink]
Show Tags
28 May 2014, 21:42
5
This post received KUDOS
3
This post was BOOKMARKED
Answer = A = 31 \(2^{100} = 2^{10*10}\) \(= (2^{10})^{10}\) \(= 1024^{10}\) \(= (1000 + 24)^{10}\) \(= (10^3 + 24)^{10}\) \(= 10^{30} + 10 * 10^{29} * 24 + ...........\) Answer should be slight greater than 30 = 31 Answer = A
Attachments
po.jpg [ 20.96 KiB  Viewed 33072 times ]
_________________
Kindly press "+1 Kudos" to appreciate



Intern
Joined: 29 Mar 2015
Posts: 3

How many digits 2^100 has? [#permalink]
Show Tags
07 Aug 2015, 05:47
2^2=1 digit 2^3=1 digit 2^4=1 digit 2^5=2 digit 2^6=2 digit 2^7=3 digit 2^8=3 digit 2^9=3 digit 2^10=4 digits  so 2^20 will have 7 digits which is 3 digits difference > 2^30 will have 10 digits and so on.. so 2^100 will have 31 digits



Manager
Joined: 24 May 2014
Posts: 96
Location: India
GRE 1: 310 Q159 V151 GRE 2: 312 Q159 V153
GPA: 2.9

Re: How many digits 2^100 has? [#permalink]
Show Tags
28 Sep 2015, 17:47
1
This post received KUDOS
PareshGmatCan you elaborate as to how (10^3+24)^10 equates to the last step.



Current Student
Joined: 20 Mar 2014
Posts: 2644
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: How many digits 2^100 has? [#permalink]
Show Tags
28 Sep 2015, 18:44
1
This post received KUDOS
narendran1990 wrote: PareshGmatCan you elaborate as to how (10^3+24)^10 equates to the last step. Let me try to explain. \((10^3+24)^{10}\) , by binomial expansion of \((a+b)^n\) = \(a^n+a^{n1}*b......+a*b^{n1}+b^n\) Similarly, \((10^3+24)^{10} = 10^{30}+....+10*24^9+24^{10}\) \(10^{29}*24+.....+24^{10}\) will have digits of the same order as that for \(10^{30}\) to reach a total of 31 digits. Options CE are way too much for this question and hence are easily eliminated.



Intern
Joined: 24 Sep 2015
Posts: 4

Re: How many digits 2^100 has? [#permalink]
Show Tags
29 Sep 2015, 00:31
5
This post received KUDOS
4
This post was BOOKMARKED
sparky wrote: How many digits 2^100 has?
A) 31 B) 35 C) 50 D) 99 E) 101 The characteristic of the logarithm of a positive number is positive and it is one less than the number of digits in the number. Take logarithm of the given number: =log(2^100) =100*log(2) =100*0.301 =30.1 hence total number of digits: =30+1 31



Intern
Joined: 21 Mar 2014
Posts: 38

Re: How many digits 2^100 has? [#permalink]
Show Tags
19 Oct 2015, 06:17
aishwarya276981 wrote: 2^2=1 digit 2^3=1 digit 2^4=1 digit 2^5=2 digit 2^6=2 digit 2^7=3 digit 2^8=3 digit 2^9=3 digit 2^10=4 digits  so 2^20 will have 7 digits which is 3 digits difference > 2^30 will have 10 digits and so on.. so 2^100 will have 31 digits 2^4 =16 it has a cycle of three.... i.e after every 3 powers, a digit increase ... but alas! it doesnt provide the correct answer
_________________
kinaare paaon phailane lage hian, nadi se roz mitti kat rahi hai....



Intern
Joined: 21 Mar 2014
Posts: 38

Re: How many digits 2^100 has? [#permalink]
Show Tags
19 Oct 2015, 06:21
Bunuel 2^1 = 1 digit 2^2 = 1 digit 2^3 = 1 digit 2^4, 2^5, 2^6 has 2 digits 2^7, 2^8, 2^9 has 3 digits there is a repetition of 3. is there a way to solve this question through this approach? my answer comes out to be 34 though.... 100/3 +1
_________________
kinaare paaon phailane lage hian, nadi se roz mitti kat rahi hai....



Current Student
Joined: 20 Mar 2014
Posts: 2644
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

How many digits 2^100 has? [#permalink]
Show Tags
19 Oct 2015, 06:49
1
This post received KUDOS
1
This post was BOOKMARKED
saroshgilani wrote: Bunuel 2^1 = 1 digit 2^2 = 1 digit 2^3 = 1 digit 2^4, 2^5, 2^6 has 2 digits 2^7, 2^8, 2^9 has 3 digits there is a repetition of 3. is there a way to solve this question through this approach? my answer comes out to be 34 though.... 100/3 +1 The mistake you are doing is that you are assuming that the 'cyclicity' is actually 3. This is wrong 2^6 = 2 digits, 2^7 to 2^9= 3 digits BUT 2^10 to 2^14 = 4 digits (this breaks the cyclicity). Thus, you can not use cyclicity for this question. Easiest way is to use log function as mentioned above (but GMAT doesnt want you to know how log function works. If you do know it, becomes very straightforward). If not, use binomial theorem as mentioned in my post howmanydigits2100has17192.html#p1579232\ Alternately, you can solve it as : 2^100 = (2^10)^10 = 1024^10 = (1.024)^10* (1000)^10 = (a value just slightly greater than 1)*10^30, giving you a number with 31 digits. Hope this helps.



Manager
Joined: 23 Sep 2015
Posts: 87
Concentration: General Management, Finance
GMAT 1: 680 Q46 V38 GMAT 2: 690 Q47 V38
GPA: 3.5

Re: How many digits 2^100 has? [#permalink]
Show Tags
27 Oct 2015, 12:17
4
This post was BOOKMARKED
yeah i used the log function for this as well.
Very quick and easy
quickly Log2^100 is 100Log2 = 100* Log(2)
I would remember a few common logs in case a question like this pops up
Log(2) = 0.305 Log(3)=0.477 Log(4)=Log(4^2)=2Log(2) Log(5)=0.698 Log(6)=Log(2*3)=Log(2)=Log(3) Log(7)=0.845
Now, as we can seem any simple common base we can use this trick for, unless they give us something a bit more complicated like 13 or 17, then I am out of tricks. But this is good for numerous ways this question can be asked.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8079
Location: Pune, India

Re: How many digits 2^100 has? [#permalink]
Show Tags
28 Oct 2015, 23:35
saroshgilani wrote: Bunuel 2^1 = 1 digit 2^2 = 1 digit 2^3 = 1 digit 2^4, 2^5, 2^6 has 2 digits 2^7, 2^8, 2^9 has 3 digits there is a repetition of 3. is there a way to solve this question through this approach? my answer comes out to be 34 though.... 100/3 +1 First of all, your AP is 1, 4, 7, 10, 13 ... 100. So number of terms will be (100  1)/3 + 1 = 34. It will not be 100/3 + 1 Next what you need to understand here is that the 34 gives you the maximum number of digits that \(2^{100}\) can have. Look at it this way. 128 > 256 > 512 > 4 digit number But this is the quickest that the number of digits can change. For some digits, you could actually have 4 numbers. 1024 > 2048 > 4096 >8192 > 5 digit number Fortunately, the only option that is less than 34 (but more than 25) is 31. Hence it must be the answer.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Intern
Joined: 13 Jan 2016
Posts: 2

How many digits 2^100 has? [#permalink]
Show Tags
27 Jan 2016, 21:30
Looking through the solutions, logarithms appear to be the fastest and most accurate way to solve this problem assuming you've memorized some basic logarithms. I haven't been a member long enough or posted enough to include a link but if you search youtube for a video titled "Logarithms Example  4 / Find The Number Of Digits using Logarithms  Maths Arithmetic" you can see a good example of using logarithms do determine the number of digits in an exponent that would take far too long to compute manually.



Manager
Joined: 03 Jan 2015
Posts: 84

Re: How many digits 2^100 has? [#permalink]
Show Tags
30 Mar 2016, 08:21
I think VeritasPrepKrisma's method is the easiest to follow, especially for those (like me) who have no idea what logarithms are.



Director
Affiliations: GMATQuantum
Joined: 19 Apr 2009
Posts: 605

Re: How many digits 2^100 has? [#permalink]
Show Tags
15 Apr 2016, 22:03
I don't think this question is something that the GMAT writers will test on the exam, unless someone here can confirm that such a question has appeared on the official GMAT. To the best of my knowledge, it does not follow the rules of how and what GMAT test writers like to test.
Here is an example of a question that is tested on the GMAT:
The number of digits in \((8^{16}) (25^{25})\) (when written in the decimal form) is .....
Cheers, Dabral



Intern
Joined: 13 Sep 2015
Posts: 17

Re: How many digits 2^100 has? [#permalink]
Show Tags
26 Aug 2016, 07:25
First of all, your AP is 1, 4, 7, 10, 13 ... 100. So number of terms will be (100  1)/3 + 1 = 34. It will not be 100/3 + 1
Karishma kindly explain this step.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8079
Location: Pune, India

Re: How many digits 2^100 has? [#permalink]
Show Tags
29 Aug 2016, 02:22
deeksha6 wrote: First of all, your AP is 1, 4, 7, 10, 13 ... 100. So number of terms will be (100  1)/3 + 1 = 34. It will not be 100/3 + 1
Karishma kindly explain this step. How do you find the number of terms in an AP? Number of terms (n) = (Last term  First term)/Common difference + 1 It is derived from Last term = First term + (n  1) * Common difference So it will be n = (100  1)/3 + 1 = 34 Check out this post for more on AP formulas: http://www.veritasprep.com/blog/2012/03 ... gressions/
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Senior Manager
Joined: 31 Jul 2017
Posts: 342
Location: Malaysia
WE: Consulting (Energy and Utilities)

How many digits 2^100 has? [#permalink]
Show Tags
31 Jan 2018, 19:52
sparky wrote: How many digits 2^100 has?
A) 31 B) 35 C) 50 D) 99 E) 101 I solved it this way  \(2^{100} = (1000 + 24)^{10}\).... Now, when you see the first term i.e. 1000 and its power 10.. the answer should lie close to 30.... 35 isn't possible because \(24^{10}\) will always be less than \(1000^{10}\).
_________________
If my Post helps you in Gaining Knowledge, Help me with KUDOS.. !!




How many digits 2^100 has?
[#permalink]
31 Jan 2018, 19:52






