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

It is currently 19 Oct 2018, 13:27

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 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6

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

Hide Tags

Manager
Manager
avatar
G
Joined: 12 Jun 2016
Posts: 217
Location: India
Concentration: Technology, Leadership
WE: Sales (Telecommunications)
GMAT ToolKit User
What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 06 Jul 2016, 08:07
2
9
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

65% (01:54) correct 35% (02:10) wrong based on 216 sessions

HideShow timer Statistics

What is the remainder when \(8^1+ 8^2+ 8^3……+8^{15}\) is divided by 6

A. 0
B. 1
C. 2
D. 4
E. 5

_________________

My Best is yet to come!

Senior Manager
Senior Manager
avatar
Joined: 02 Mar 2012
Posts: 317
Schools: Schulich '16
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 06 Jul 2016, 23:52
1
C

8/6 has remendar 2 and 64/6 has remainder 4

so there will be 8 2's and 7 4's from the sum which when added give 44 as sum.So 44/6=2 remainder
Manager
Manager
avatar
G
Joined: 12 Jun 2016
Posts: 217
Location: India
Concentration: Technology, Leadership
WE: Sales (Telecommunications)
GMAT ToolKit User
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 07 Jul 2016, 08:14
hsbinfy wrote:
C

8/6 has remendar 2 and 64/6 has remainder 4

so there will be 8 2's and 7 4's from the sum which when added give 44 as sum.So 44/6=2 remainder


hsbinfy

Thank you for the reply.

I followed you till getting the sum as 44 (This is sum of individual remainders). But I could not follow the highlighted part. Can you please say why we need to divide the sum of the remainders again by 6?

Thanks in advance!
_________________

My Best is yet to come!

Intern
Intern
avatar
Joined: 30 Jun 2016
Posts: 3
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 09 Jul 2016, 10:24
2
Hello, I approached this question with using cyclisity properties, but i don't know whether it is correct. Maybe someone can prove confirm it mathematically.

So we need to determine the remainder of the sum of the factors of 8^1 till 8^15

I determined the cycle of 8 and got

8^1 = 8
8^2 = 64
8^3 = 512
8^4 = 4096
8^5 = 32768

So I just added all the ones digits we will get if we sum 8^1 to 8^15, so we would get [(8+4+2+6)*3 + 8+4+2]/6 = 74/6 = 12 with remainder = 2.

Can anyone tell me if it is correct or coincidence?
Manager
Manager
avatar
Joined: 01 Jan 2015
Posts: 63
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 09 Jul 2016, 11:29
Matthias1205 wrote:
Hello, I approached this question with using cyclisity properties, but i don't know whether it is correct. Maybe someone can prove confirm it mathematically.

So we need to determine the remainder of the sum of the factors of 8^1 till 8^15

I determined the cycle of 8 and got

8^1 = 8
8^2 = 64
8^3 = 512
8^4 = 4096
8^5 = 32768

So I just added all the ones digits we will get if we sum 8^1 to 8^15, so we would get [(8+4+2+6)*3 + 8+4+2]/6 = 74/6 = 12 with remainder = 2.

Can anyone tell me if it is correct or coincidence?


No, it is not correct. See my post: http://gmatclub.com/forum/what-is-the-remainder-when-207423.html#p1591521
Current Student
User avatar
B
Status: DONE!
Joined: 05 Sep 2016
Posts: 384
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 09 Nov 2016, 15:11
1
1
You'll notice a pattern with the remainder if you divide 8 by 6 first, 64/8 and so on....

Remainder will jump back and forth between 2 (on odd exponents) and 4 (on even exponents)

If you count the number of remainder 2's and 4's --> 8 2's & 7 4's--> 2(8)+7(4) = 16+28 = 44

44/6 = 2 R2
Director
Director
User avatar
G
Joined: 23 Jan 2013
Posts: 575
Schools: Cambridge'16
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 12 Nov 2016, 06:24
cyclicity of remainder is

8^1=2
8^2=4
8^3=2
8^4=4
all odd degrees of 8 has remainder 2 and all even has 4. So 8^15 has remainder 2

C
Board of Directors
User avatar
P
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4094
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 12 Nov 2016, 12:29
3
1
susheelh wrote:
What is the remainder when \(8^1+ 8^2+ 8^3……+8^{15}\) is divided by 6

A. 0
B. 1
C. 2
D. 4
E. 5


\(\frac{8}{6}\) = Remainder 2

\(\frac{8^2}{6}\) = Remainder 4

\(\frac{8^3}{6}\) = Remainder 2

\(\frac{8^4}{6}\) = Remainder 4


Thus odd powers will have remainder 2 ; and even powers will have remainder 4

Now, \(8^1+ 8^2+ 8^3……+8^{15}\) will have the following powers -

Odd = 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 ( 8 odd powers ) ; Sum of remainder = 16

Even Powers = 2, 4 , 6 , 8 , 10 , 12 , 14 ( 7 even Powers ) ; Sum of remainder = 28

Total sum of remainder = 44

44/6 = Remainder 2

Hence correct aswer will be (C) 2....

_________________

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 )

Senior Manager
Senior Manager
avatar
B
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 03 Dec 2016, 07:37
susheelh wrote:
What is the remainder when \(8^1+ 8^2+ 8^3……+8^{15}\) is divided by 6

A. 0
B. 1
C. 2
D. 4
E. 5


\(8 = 2\) (mod 6) ---> “leaves remainder 2 when divided by 6”.

In mod 6 our expression equals to:

\(\frac{2^1 + 2^2 + 2^3 + … + 2^{15}}{6} = \frac{2*(2^{15}-1)}{(2 – 1)*6} = \frac{2*(2^{15} – 1)}{2*3} = \frac{2^{15} – 1}{3}\)

Now \(2 = -1\) (mod 3) and we have

\(\frac{(-1)^{15} – 1}{3} = \frac{-1 – 1}{3} = \frac{-2}{3}\)

Remainder -2 is the same as remainder 1 when divided by 3 but we need to multiply by cancelled factor 2

\(\frac{1}{3} = \frac{2}{6}\)

Our remainder is \(2\).

Answer C.
Current Student
User avatar
D
Joined: 12 Aug 2015
Posts: 2638
Schools: Boston U '20 (M)
GRE 1: Q169 V154
GMAT ToolKit User Premium Member
What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 04 Dec 2016, 00:34
Here is my Take on this Question=>
USE PATTERN RECOGNITION

In Questions such as the one above,observing the pattern can be really helpful/
8=> k=6k+2
8+64=>6k
8+64+512=> 6k+2
8+64+512+4096=>6k
Hence for the number of terms being odd => Remainder is 2 and for the number of terms being even => Remainder is 0.
Since 15=odd
The remainder must be 2

Hence C

_________________


MBA Financing:- INDIAN PUBLIC BANKS vs PRODIGY FINANCE!

Getting into HOLLYWOOD with an MBA!

The MOST AFFORDABLE MBA programs!

STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+)

AVERAGE GRE Scores At The Top Business Schools!

Manager
Manager
avatar
S
Joined: 30 Mar 2017
Posts: 136
GMAT 1: 200 Q1 V1
Reviews Badge
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6  [#permalink]

Show Tags

New post 05 Aug 2018, 17:58
susheelh wrote:
What is the remainder when \(8^1+ 8^2+ 8^3……+8^{15}\) is divided by 6

A. 0
B. 1
C. 2
D. 4
E. 5


\(8^1+ 8^2+ 8^3……+8^{15}\) is same as
\(2^3+ 2^6+ 2^9……+2^{45}\)
What is the remainder when expression above is divided by 6?

To simplify, what is the remainder of
\(2^2+ 2^5+ 2^8……+2^{44}\)
when divided by 3? (Once we find the remainder, remember to multiply it by 2. Since we're factoring out 2 to make the expression easier to work with, we have to re-insert the factor of 2 at the end.)

Pattern:
1st term: r=1
2nd term: r=2
3rd term: r=1
etc
This means that the sum of every 2 terms will have r=0 (when div by 3). So, if there is an even number of terms, r=0; if there is an odd number of terms, r=1.

Total # of terms: \(\frac{44-2}{3}+1=15\)
So r=1 when div by 3. Re-introduce the 2 we factored out earlier, and r=2 when div by 6.

Answer: C
GMAT Club Bot
Re: What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6 &nbs [#permalink] 05 Aug 2018, 17:58
Display posts from previous: Sort by

What is the remainder when 8^1+ 8^2+ 8^3……. + 8^15 is divided by 6

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


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®.