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

It is currently 24 Sep 2018, 22:21

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

What is the remainder when 5^68 is divided by 7?

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

Hide Tags

Intern
Intern
User avatar
B
Joined: 07 Dec 2016
Posts: 42
Reviews Badge
What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 14 Apr 2017, 05:26
3
16
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

48% (01:57) correct 52% (02:02) wrong based on 338 sessions

HideShow timer Statistics

What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6

_________________

Cheers!
If u like my post..... payback in Kudos!! :beer

Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 6808
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 14 Apr 2017, 07:47
3
4
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


Hi..

Make use of expansion ...
\(5^{68}=(7-2)^{68}\)
When you expand the equation, all terms will be div by 7 except
\((-2)^{68}=2^{68}=(2^3)^{22}*2^2=8^{22}*4=(7-1)^{22}*4\)..
In (7-1)^22 all terms will be div by 7 except (-1)^22, which is same as 1..
So remainder is 1*4=4
D
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Most Helpful Community Reply
Board of Directors
User avatar
P
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4033
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 14 Apr 2017, 07:45
4
4
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


\(\frac{5^{68}}{7} = \frac{( 7 - 2 )^{68}}{7}\)

\(\frac{2^1}{7}\)= Remainder 2
\(\frac{2^2}{7}\)= Remainder 4
\(\frac{2^3}{7}\)= Remainder 1

\(\frac{2^4}{7}\)= Remainder 2
\(\frac{2^5}{7}\)= Remainder 4
\(\frac{2^6}{7}\)= Remainder 1

So, We have a pattern here, the cyclicity is of 3.

\(\frac{68}{3}\) = Remainder is 2

Thus, the remainder corresponding to the value will be 4, answer must be (D) 4
_________________

Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )

General Discussion
Manager
Manager
avatar
G
Joined: 27 Dec 2016
Posts: 224
CAT Tests
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 11 May 2017, 12:30
Hi,

Can anyone please tell me why we did (7-2) at the beginning and not (8-3) or some other number? What is the concept behind this? Would be greatly appreciated if someone could please explain me the reasoning and concept behind this trick!

Thanks!
Senior CR Moderator
User avatar
V
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1385
Location: Viet Nam
GMAT ToolKit User Premium Member
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 12 May 2017, 01:19
1
1
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


\(\frac{5^{68}}{7}=\frac{(-2)^{68}}{7}=\frac{2^{68}}{7}=\frac{(2^3)^{22}\times 2^2}{7}
=\frac{8^{22} \times 2^2}{7}=\frac{1\times 2^2}{7}=4\)

The answer is D.
_________________

Actual LSAT CR bank by Broall

How to solve quadratic equations - Factor quadratic equations
Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

New Error Log with Timer

Senior CR Moderator
User avatar
V
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1385
Location: Viet Nam
GMAT ToolKit User Premium Member
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 12 May 2017, 01:23
csaluja wrote:
Hi,

Can anyone please tell me why we did (7-2) at the beginning and not (8-3) or some other number? What is the concept behind this? Would be greatly appreciated if someone could please explain me the reasoning and concept behind this trick!

Thanks!


The reason is that we need to find the remainder when dividing by 7. Hence 5=7-2.

Also, when we extract the expression \((7-2)^n\) we will have an expression like \(7^k \times A + (-2)^n\) with \(k\) is an positive integer.

Note that \(7^k \times A\) is divisible by 7, so we simply find the remainder of \((-2)^n\) when dividing by 7.

Hence, we won't do like this way 5=8-3 since 8 isn't divisible by 7 and this method makes the problem more complex.
_________________

Actual LSAT CR bank by Broall

How to solve quadratic equations - Factor quadratic equations
Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

New Error Log with Timer

Director
Director
User avatar
P
Joined: 13 Mar 2017
Posts: 619
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 12 May 2017, 03:16
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


We can do it by expanding the number .. {} remainder value in the below explanation
{5^68 / 7}= {25^34 /7} = {4^34 /7} = {16^17 /7 } ={2^17/7} = {((2^3)^5 * 2^2) /7} = {((8)^5 * 2^2) /7} = 1 * 4 = 4

So Remainder is 4.
_________________

CAT 99th percentiler : VA 97.27 | DI-LR 96.84 | QA 98.04 | OA 98.95
UPSC Aspirants : Get my app UPSC Important News Reader from Play store.

MBA Social Network : WebMaggu


Appreciate by Clicking +1 Kudos ( Lets be more generous friends.)



What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish".

Director
Director
User avatar
P
Joined: 13 Mar 2017
Posts: 619
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 12 May 2017, 03:27
shashankism wrote:
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


We can do it by expanding the number .. {} remainder value in the below explanation
{5^68 / 7}= {25^34 /7} = {4^34 /7} = {16^17 /7 } ={2^17/7} = {((2^3)^5 * 2^2) /7} = {((8)^5 * 2^2) /7} = 1 * 4 = 4

So Remainder is 4.


This can also be solved by following method .. Considering -ve values in expansion.

{5^68 / 7} = {-2^68 /7} = {2^68 /7 } = {8^22*2^2/7} = { 4/7 } = 4
_________________

CAT 99th percentiler : VA 97.27 | DI-LR 96.84 | QA 98.04 | OA 98.95
UPSC Aspirants : Get my app UPSC Important News Reader from Play store.

MBA Social Network : WebMaggu


Appreciate by Clicking +1 Kudos ( Lets be more generous friends.)



What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish".

Intern
Intern
avatar
Joined: 13 Mar 2016
Posts: 1
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 15 May 2017, 10:21
The expansion can be written as (126-1)^22 *5^2 divided by 7.

This approach would be very simple and easy to solve this type of questions;

Take 5^3 = 125 and restructure the question like this : (5^3)^22 * 5^2 divided by 7.

The main idea is to expand the numerator with a number close to 7 or multiples of 7 by 1.

If I take 5^3 =125 = 126-1. Here 126 is divisible by 7 or a multiple of 7.

Now it becomes very simple. If we expand this (126-1)^22 using binomial expansion, the remainder term would be (-1)^22, which is 1.

So we left with 5^2 divided by 7 which is 25/7 and the remainder is 4.

Hence always try to expand the numerator such that we convert a big part of the problem into 1 and then find the remainder on the remaining part.

Thanks
Manager
Manager
avatar
B
Joined: 29 May 2017
Posts: 56
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 12 Sep 2018, 07:54
chetan2u wrote:
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


Hi..

Make use of expansion ...
\(5^{68}=(7-2)^{68}\)
When you expand the equation, all terms will be div by 7 except
\((-2)^{68}=2^{68}=(2^3)^{22}*2^2=8^{22}*4=(7-1)^{22}*4\)..
In (7-1)^22 all terms will be div by 7 except (-1)^22, which is same as 1..
So remainder is 1*4=4
D


Hi Chetan.....can you tel me if the method below will also work?

5^68/7 equals (125^22 x 5^2)/7
thus we get a remainder of -1 for 125^22, which means:
(-1)^22 x 25 -> 1 x 25 -> 25 -> 25/7 -> remainder is 4.

thanks
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 6808
Re: What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 12 Sep 2018, 07:58
Mansoor50 wrote:
chetan2u wrote:
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


Hi..

Make use of expansion ...
\(5^{68}=(7-2)^{68}\)
When you expand the equation, all terms will be div by 7 except
\((-2)^{68}=2^{68}=(2^3)^{22}*2^2=8^{22}*4=(7-1)^{22}*4\)..
In (7-1)^22 all terms will be div by 7 except (-1)^22, which is same as 1..
So remainder is 1*4=4
D


Hi Chetan.....can you tel me if the method below will also work?

5^68/7 equals (125^22 x 5^2)/7
thus we get a remainder of -1 for 125^22, which means:
(-1)^22 x 25 -> 1 x 25 -> 25 -> 25/7 -> remainder is 4.

thanks


Yes Mansoor, you are absolutely correct with your approach
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Manager
Manager
User avatar
S
Joined: 19 Feb 2010
Posts: 166
What is the remainder when 5^68 is divided by 7?  [#permalink]

Show Tags

New post 16 Sep 2018, 12:45
amathews wrote:
What is the remainder when 5^68 is divided by 7?

A) 1
B) 2
C) 3
D) 4
E) 6


Question is 5^68%7 = ?

This can be written as (125)^22 x 5^2

125% 7 = 6 or -1
So, 125^22%7 = -1^22 = 1
25%7 = 4
Hence, the remainder is 1x4 = 4
D.

To understand a smart approach for solving such questions in 1-minute, please watch the following video; the concept has been explained in ~3 minutes.



Hope this helps.

All the best!
Experts' Global Team
_________________

GMAT Preparation Online- World's Most Exhaustive GMAT Prep Program | Gain 7-days demo access (includes a full length test)

GMAT Club Bot
What is the remainder when 5^68 is divided by 7? &nbs [#permalink] 16 Sep 2018, 12:45
Display posts from previous: Sort by

What is the remainder when 5^68 is divided by 7?

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

Events & Promotions

PREV
NEXT


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| 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®.