It is currently 24 Nov 2017, 06:22

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 3^123 is divided by 5?

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

Hide Tags

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42356

Kudos [?]: 133204 [0], given: 12439

What is the remainder when 3^123 is divided by 5? [#permalink]

Show Tags

New post 27 Apr 2016, 11:23
Expert's post
5
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  5% (low)

Question Stats:

78% (00:42) correct 22% (00:43) wrong based on 142 sessions

HideShow timer Statistics

Kudos [?]: 133204 [0], given: 12439

Manager
Manager
avatar
B
Joined: 16 Feb 2016
Posts: 53

Kudos [?]: 20 [0], given: 26

Concentration: Other, Other
GMAT ToolKit User Premium Member Reviews Badge
Re: What is the remainder when 3^123 is divided by 5? [#permalink]

Show Tags

New post 27 Apr 2016, 20:18
Ok lets try this one.

\(\ 3^{123}= \ (5-2)^{123}= \ 5^{123}+(-2)^{123}\)

\(\ 5^{123}\) is always divisible by 5.

So need to look at the \(\ -2^{123}\)

Units digit of -2 to the power of n is cyclical and consists of 4 possibilities {-2, 4, -8, 6}

Hence \(\ (-2)^{120}\) will have last digit 6 and \(\ (-2)^{123}\) = -8.

Dividing -8 by 5 we have remainder

-8=5n+R
For n=-2
-8=5*-2+R

R=2
Hence Answer is (C)

Kudos [?]: 20 [0], given: 26

Expert Post
Math Expert
User avatar
P
Joined: 02 Aug 2009
Posts: 5234

Kudos [?]: 5894 [0], given: 118

Re: What is the remainder when 3^123 is divided by 5? [#permalink]

Show Tags

New post 27 Apr 2016, 20:33
maipenrai wrote:
Ok lets try this one.

\(\ 3^{123}= \ (5-2)^{123}= \ 5^{123}+(-2)^{123}\)

\(\ 5^{123}\) is always divisible by 5.

So need to look at the \(\ -2^{123}\)

Units digit of -2 to the power of n is cyclical and consists of 4 possibilities (-2, 4, -8, 6)

Hence \(\ (-2)^{120}\) will have last digit 6 and \(\ (-2)^{123}\) = -8.

Dividing 8 by -5 we have remainder

8=-5n+R
For n=2
8=-5*2+R

R=2
Hence Answer is (C)


hi,
another approach after the highlighted portion
So need to look at the \(\ -2^{123}\)
\(\ -2^{123}= (-2^4)^{\frac{123}{4}} = 16^{30}*-2^3=(15+1)^{30}*-8\)
now\((15+1)^{30}\) will leave 1 as remainder, so total remainder = 1*-8 = -8..
as correctly done, we cannot have negative remainder so remainder = 10-8 = 2
_________________

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

Kudos [?]: 5894 [0], given: 118

1 KUDOS received
Intern
Intern
avatar
Joined: 22 Jan 2016
Posts: 7

Kudos [?]: 9 [1], given: 0

Location: Viet Nam
GMAT 1: 710 Q50 V35
GPA: 3
WE: Other (Internet and New Media)
Re: What is the remainder when 3^123 is divided by 5? [#permalink]

Show Tags

New post 27 Apr 2016, 23:34
1
This post received
KUDOS
1
This post was
BOOKMARKED
I believe there is an easier way to do as follow:
3^1 = 3; 3^2 = 9, 3^3 = 7, 3^4 = 1; So basiccally 3 has a cycle of 4.
Therefore 123: 4 = 30 with remainder of 3.
3^3 will have the unit digit of 7, thereby dividing by 5 will leave a remainder of 2.

C.

Kudos [?]: 9 [1], given: 0

Manager
Manager
avatar
B
Joined: 16 Feb 2016
Posts: 53

Kudos [?]: 20 [0], given: 26

Concentration: Other, Other
GMAT ToolKit User Premium Member Reviews Badge
Re: What is the remainder when 3^123 is divided by 5? [#permalink]

Show Tags

New post 28 Apr 2016, 22:56
1
This post was
BOOKMARKED
gnahanut wrote:
I believe there is an easier way to do as follow:
3^1 = 3; 3^2 = 9, 3^3 = 7, 3^4 = 1; So basiccally 3 has a cycle of 4.
Therefore 123: 4 = 30 with remainder of 3.
3^3 will have the unit digit of 7, thereby dividing by 5 will leave a remainder of 2.

C.


Simple and straight to the point!

Kudos [?]: 20 [0], given: 26

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15499

Kudos [?]: 283 [0], given: 0

Premium Member
Re: What is the remainder when 3^123 is divided by 5? [#permalink]

Show Tags

New post 26 Sep 2017, 08:07
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 [?]: 283 [0], given: 0

Re: What is the remainder when 3^123 is divided by 5?   [#permalink] 26 Sep 2017, 08:07
Display posts from previous: Sort by

What is the remainder when 3^123 is divided by 5?

  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®.