It is currently 20 Jan 2018, 03:25

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

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

What is the remainder when divide 2^100 by 100?

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

Hide Tags

1 KUDOS received
Senior CR Moderator
User avatar
V
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1317

Kudos [?]: 1188 [1], given: 62

Location: Viet Nam
GMAT ToolKit User Premium Member CAT Tests
What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 23 Nov 2016, 08:01
1
This post received
KUDOS
6
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

45% (02:53) correct 55% (02:24) wrong based on 64 sessions

HideShow timer Statistics


Last edited by broall on 23 Nov 2016, 15:45, edited 1 time in total.

Kudos [?]: 1188 [1], given: 62

1 KUDOS received
Senior Manager
Senior Manager
User avatar
P
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 429

Kudos [?]: 158 [1], given: 423

Location: India
GPA: 3.64
GMAT ToolKit User Premium Member Reviews Badge
Re: What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 23 Nov 2016, 09:51
1
This post received
KUDOS
1
This post was
BOOKMARKED
We need the last two digits of 2^100.
Since 100 is divisible by 4, by the cyclicity of 2, last digit must be 6.
2^4=16
2^8=256 (difference of 56-16=40)
2^12=4096 (difference of 96-56=40)
2^16=___36 (last 2 digits)
2^20=___76 (last 2 digits)
2^24=___16
Observe that last 2 digits start repeating.
Hence 2^100 must have last 2 digits as 76 since 100 is divisible by 5 and the series repeats after 5 terms.
Answer D

If u liked my post, please press kudos!
_________________

Please give kudos, if you like my post

When the going gets tough, the tough gets going...

Kudos [?]: 158 [1], given: 423

Senior Manager
Senior Manager
avatar
B
Joined: 13 Oct 2016
Posts: 367

Kudos [?]: 415 [0], given: 40

GPA: 3.98
Re: What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 23 Nov 2016, 11:50
\(\frac{2^{100}}{100} = \frac{2^{100}}{4*25}\)

We’ll find a remainder for 4 and 25 separately and use a Chinese remainder theorem to find a number.

\(2^{100} = 0 (mod 4)\) so our \(N = 0 (mod 4)\)

For 25 we’ll use slightly different approach.

GCF of \(2^{100}\) and \(25\) is \(1\), they are co-prime and we can apply Euiler's theorem.

\(2^{20} = 1 (mod 25)\) ---> \(2^{100} = (2^{20})^5 = 1^5 = 1\)

\(N = 1 (mod 25)\)

Combining two equations we’ll get

\(25x + 1 = 4y\)

or \(25x + 1 = 0 (mod 4)\) Solving this linear congruence we’ll get \(x=3 (mod 4)\)

and our number \(N = 25*3+1 =76\)

Answer D.

Kudos [?]: 415 [0], given: 40

Senior Manager
Senior Manager
avatar
B
Joined: 13 Oct 2016
Posts: 367

Kudos [?]: 415 [0], given: 40

GPA: 3.98
Re: What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 23 Nov 2016, 12:24
2
This post was
BOOKMARKED
nguyendinhtuong wrote:
What is the remainder when divide \(2^{100}\) by 100?

A. 16
B. 36
C. 56
D. 76
E. 96


Another approach

\(\frac{2^{100}}{100} = \frac{2^{100}}{2^2*25} = \frac{2^{98}}{25}\)

We know that \(2^7 = 128 = 3 (mod 25)\)

\(\frac{2^{98}}{25} = \frac{(2^7)^{14}}{25} = \frac{3^{14}}{25}\)

This will not help us much, but \(3^3 = 27 = 2 (mod 25)\)

And we have:

\(\frac{(3^3)^4*3^2}{25} = \frac{2^4*3^2}{25} = \frac{144}{25} = -6 (mod 25)\)

Now we need to multiply by cancelled factor \(2^2\)

\(-6*4 (mod 25*4) = -24 (mod 100) = 76 (mod 100)\)

Our remainder is 76

Kudos [?]: 415 [0], given: 40

1 KUDOS received
Senior Manager
Senior Manager
avatar
B
Joined: 13 Oct 2016
Posts: 367

Kudos [?]: 415 [1], given: 40

GPA: 3.98
Re: What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 23 Nov 2016, 13:42
1
This post received
KUDOS
2
This post was
BOOKMARKED
Approach #3 very fast

\(\frac{2^{100}}{100}\) means we need to find last two digits of \(2^{100}\)

We know that \(2^{10}=1024\)

\(24^{odd}\) last 2 digits = \(24\)

\(24^{even}\) last 2 digits = \(76\)

\(2^{100}\) = \((2^{10})^{10}\) = \((1024)^{10}\) = \(24^{even}\)

Last two digits are 76

Answer D.

Kudos [?]: 415 [1], given: 40

Expert Post
Math Expert
User avatar
D
Joined: 02 Aug 2009
Posts: 5534

Kudos [?]: 6439 [0], given: 122

Re: What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 23 Nov 2016, 18:10
nguyendinhtuong wrote:
What is the remainder when divide \(2^{100}\) by 100?

A. 16
B. 36
C. 56
D. 76
E. 96



Hi,
One approach that can be used in all such Qs is to find closest multiples taking power..
Example here itself
\(2^100= (2^9)^{11}*2=512^{11}*2\)...
Now 512 to some power will leave a remainder same as 12 to that power..
So the remainder will be same as \(12^{11}*2\)...
\(12^{11}*2=(10+2)^{11}*2\)
Now when you expand this all terms except two terms willbe div by 100..
\(2^{11}*10^0+2^{10}*10^1=2048+1024*10\)
Last two digits are 48+40=88..
Remainder=88*2=176
That is 76
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


BANGALORE/-

Kudos [?]: 6439 [0], given: 122

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 14223

Kudos [?]: 291 [0], given: 0

Premium Member
Re: What is the remainder when divide 2^100 by 100? [#permalink]

Show Tags

New post 04 Dec 2017, 03:22
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 Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 291 [0], given: 0

Re: What is the remainder when divide 2^100 by 100?   [#permalink] 04 Dec 2017, 03:22
Display posts from previous: Sort by

What is the remainder when divide 2^100 by 100?

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


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

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

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®.