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

It is currently 17 Feb 2019, 18:12

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 in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
  • Free GMAT Algebra Webinar

     February 17, 2019

     February 17, 2019

     07:00 AM PST

     09:00 AM PST

    Attend this Free Algebra Webinar and learn how to master Inequalities and Absolute Value problems on GMAT.
  • Valentine's day SALE is on! 25% off.

     February 18, 2019

     February 18, 2019

     10:00 PM PST

     11:00 PM PST

    We don’t care what your relationship status this year - we love you just the way you are. AND we want you to crush the GMAT!

What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is

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

Hide Tags

 
Senior Manager
Senior Manager
avatar
Joined: 30 Aug 2003
Posts: 315
Location: BACARDIVILLE
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post Updated on: 20 Feb 2012, 22:43
3
33
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

61% (01:31) correct 39% (01:46) wrong based on 1149 sessions

HideShow timer Statistics

What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is divided by 6?

A. 0
B. 3
C. 2
D. 5
E. None of the above

_________________

Pls include reasoning along with all answer posts.
****GMAT Loco****
Este examen me conduce jodiendo loco


Originally posted by sunniboy007 on 13 Mar 2004, 15:32.
Last edited by Bunuel on 20 Feb 2012, 22:43, edited 2 times in total.
Edited the question and added the OA
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 52906
Re: What is the remainder when 9^1 + 9^2 + 9^3 +....+ 9^9 is  [#permalink]

Show Tags

New post 20 Feb 2012, 22:40
22
15
sunniboy007 wrote:
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is divided by 6?

A. 0
B. 3
C. 2
D. 5
E. None of the above


30 sec approach:
Given: \(9^1+(9^2+9^3+9^4+9^5+9^6+9^7+9^8+9^9)\). Notice that in the brackets we have the sum of 8 odd multiples of 3, hence the sum in the brackets will be even multiple of 3 (the sum of 8 odd numbers is even). So, the sum in the brackets is multiple of 6 (remainder is zero). So we are just left with the first term 9, which yields remainder of 3 upon division by 6.

Answer: B.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Most Helpful Community Reply
Manager
Manager
User avatar
Status: Employed
Joined: 17 Nov 2011
Posts: 80
Location: Pakistan
Concentration: International Business, Marketing
GMAT 1: 720 Q49 V40
GPA: 3.2
WE: Business Development (Internet and New Media)
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 21 Feb 2012, 12:50
19
6
Don't really know if my approach is correct but this is how I approached it.

When divided by \(6\), \(9^1\) leaves a remainder of \(3\)
When divided by \(6\), \(9^2\) leaves a remainder of \(3\)
When divided by \(6\), \(9^3\) leaves a remainder of \(3\)

You can check further if you want to, but at this point I had decided that all the terms individually leave a remainder of \(3\), so all the remainder added up would be \(9*3=27\) , and \(27\) divided by \(6\) leaves a remainder of \(3\) . Hence the answer should be B.

If I am correct, remainders can be added and then divided by the original number to come up with the remainder. For example, lets take two numbers, \(11\) and \(13\) and divide them by \(4\). \(11\) and \(13\) add up to \(24\) and \(24\) divided by \(4\) leaves a remainder of \(0\). \(11\) divided by \(4\) leaves a remainder of \(3\), \(13\) divided by \(4\) leaves a remainder of \(1\). Now when you add the remainders, \(3+1=4\), which leaves a remainder of 0 when divided by \(4\) or is divisible by \(4\).
_________________

"Nowadays, people know the price of everything, and the value of nothing." Oscar Wilde

General Discussion
Intern
Intern
avatar
Joined: 08 Mar 2012
Posts: 3
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 16 Mar 2012, 14:36
3
Hi.. i did it like -
the last digits are -
9^1 = 9
9^2 = 8
9^3 = 7
9^4 = 6
till
9^9 = 1

=> adding no. from 1 to 9 = 45
45/6 = 3
Manager
Manager
User avatar
Status: Writing Essays
Joined: 25 Nov 2011
Posts: 132
Location: Brazil
Concentration: Technology, Finance
GMAT 1: 720 Q44 V47
WE: Information Technology (Commercial Banking)
Reviews Badge
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 16 Mar 2012, 14:54
1
monalimishra wrote:
Hi.. i did it like -
the last digits are -
9^1 = 9
9^2 = 8
9^3 = 7
9^4 = 6
till
9^9 = 1

=> adding no. from 1 to 9 = 45
45/6 = 3


There is a flaw in your aproach... 9^2 = 81
Intern
Intern
avatar
Joined: 08 Mar 2012
Posts: 3
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 16 Mar 2012, 19:20
yup... realized my mistake sometime after posting... :-(
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 52906
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 26 Jun 2013, 01:24
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

All DS Divisibility/Multiples/Factors questions to practice: search.php?search_id=tag&tag_id=354
All PS Divisibility/Multiples/Factors questions to practice: search.php?search_id=tag&tag_id=185

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Manager
Manager
avatar
Joined: 14 Nov 2011
Posts: 120
Location: United States
Concentration: General Management, Entrepreneurship
GPA: 3.61
WE: Consulting (Manufacturing)
GMAT ToolKit User
Re: What is the remainder when 9^1 + 9^2 + 9^3 +....+ 9^9 is  [#permalink]

Show Tags

New post 12 Jul 2013, 02:45
Bunuel wrote:
sunniboy007 wrote:
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is divided by 6?

A. 0
B. 3
C. 2
D. 5
E. None of the above


30 sec approach:
Given: \(9^1+(9^2+9^3+9^4+9^5+9^6+9^7+9^8+9^9)\). Notice that in the brackets we have the sum of 8 odd multiples of 3, hence the sum in the brackets will be even multiple of 3 (the sum of 8 odd numbers is even). So, the sum in the brackets is multiple of 6 (remainder is zero). So we are just left with the first term 9, which yields remainder of 3 upon division by 6.

Answer: B.


Hi Bunnel,

I did it as below:
Sum = 9/8*(9^9-1)
Rem (s/6) = ?

9^9 - has units digit 9,
9-1 = 8/8 = 1
9^2 = 81-1 = 80/8 = 10
9^3 = 729-1 = 728/8 = 91

Sum = 9*Integer
Rem (s, 6) = 3

Here I could do this because the integral multiple of 9 in the sum is not a multiple of 6.

Can I use this method for other cases?
Intern
Intern
avatar
Joined: 03 Oct 2012
Posts: 2
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 06 Aug 2013, 09:10
1
I searched for the patrons in the digit of nine, which resulted in 1,9,1,9,1,9..... after that I summed them up which was 49. 49 divided by 6 left a remainder of 3.
Manager
Manager
avatar
Status: Joining Cranfield Sep 2014
Joined: 01 Sep 2012
Posts: 52
Concentration: Technology, General Management
GMAT 1: 530 Q50 V14
GMAT 2: 630 Q48 V29
WE: Engineering (Energy and Utilities)
Reviews Badge
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 06 Aug 2013, 10:10
6
1
6 is an even multiple of 3. When any even multiple of 3 is divided by 6, it will leave a remainder of 0. Or in other words it is perfectly divisible by 6.

On the contrary, when any odd multiple of 3 is divided by 6, it will leave a remainder of 3. For e.g when 9 an odd multiple of 3 is divided by 6, you will get a remainder of 3.

9 is an odd multiple of 3. And all powers of 9 are odd multiples of 3.
Therefore, when each of the 9 powers of 9 listed above are divided by 6, each of them will leave a remainder of 3.

The total value of the remainder = 3 + 3 + .... + 3 (9 remainders) = 27.
27 is divisible by 6. Hence, it will leave remainder as 3.

Hence, the final remainder when the expression 9^1 + 9^2 + 9^3 + .... + 9^9 is divided by 6 will be equal to '3'.
and one more point to add if the expression is 9^1+9^2+...........+9^10 is divided by 6 then the remainter will be '0'

We can generalize it further:-
if (9^1+9^2+.......9^n) if n is odd then the remainder will always be 3 and if n is even then the remainder will always be '0'.

I hope people will like this explaination and if it helps you further please give Kudos to me.
Manager
Manager
User avatar
Joined: 18 Aug 2014
Posts: 119
Location: Hong Kong
Schools: Mannheim
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 27 Jun 2015, 01:35
sunniboy007 wrote:
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is divided by 6?

A. 0
B. 3
C. 2
D. 5
E. None of the above


9^45 = (6+3)^45 .. leaves us with 3^45..which will always give us a remainder of 3 when divided by 6 (27/6, 81/6, 9/6).. final answer B..

is this approach correct ?
Manager
Manager
User avatar
Joined: 06 Jan 2015
Posts: 59
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 19 Feb 2016, 12:54
There is a flaw you are stating that (6+3)^45 = 6^45 + 3^45 which is incorrect.

LaxAvenger wrote:
sunniboy007 wrote:
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is divided by 6?

A. 0
B. 3
C. 2
D. 5
E. None of the above


9^45 = (6+3)^45 .. leaves us with 3^45..which will always give us a remainder of 3 when divided by 6 (27/6, 81/6, 9/6).. final answer B..

is this approach correct ?

_________________

"No pain, no gain

Current Student
User avatar
D
Joined: 12 Aug 2015
Posts: 2621
Schools: Boston U '20 (M)
GRE 1: Q169 V154
GMAT ToolKit User Premium Member
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 13 Mar 2016, 23:54
Current Student
User avatar
D
Joined: 12 Aug 2015
Posts: 2621
Schools: Boston U '20 (M)
GRE 1: Q169 V154
GMAT ToolKit User Premium Member
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 17 Mar 2016, 04:42
1
The easiest way here is to find he pattern
here 9^1/6=> remainder =3
9^1+9^2/6=> reminder = 0
9^1+9^2+9^3/6=> remainder =3
hence the cyclicity is 2
so the number of terms are odd => remainder =3
hence B
_________________

Give me a hell yeah ...!!!!!

MBA Dating:- B-SCHOOL with the MOST ATTRACTIVE Women
MBA Recruiting:- EMPLOYMENT AND SALARY STATISTICS AT TOP B-SCHOOLS IN THE US! (2018)
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!

Senior Manager
Senior Manager
User avatar
S
Joined: 08 Dec 2015
Posts: 292
GMAT 1: 600 Q44 V27
Reviews Badge
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 31 Mar 2016, 09:20
An arithmetics question here, isn't 9^1 + 9^2 + 9^3 +...+ 9^9 the same as 9^11? like factor all the common nines, it will give you 9^2 (nine nines) then add them to the given 9^9 and get 9^11. Does this make sense?... We are given a sum, so Im not sure this logic works..

Thank you!
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 52906
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 31 Mar 2016, 11:19
iliavko wrote:
An arithmetics question here, isn't 9^1 + 9^2 + 9^3 +...+ 9^9 the same as 9^11? like factor all the common nines, it will give you 9^2 (nine nines) then add them to the given 9^9 and get 9^11. Does this make sense?... We are given a sum, so Im not sure this logic works..

Thank you!


No, this does not make sense. Not sure how you are getting this... You CANNOT factor out 9^2 out of 9^1 + 9^2 + 9^3 +...+ 9^9, you can only factor out 9.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
B
Joined: 30 May 2017
Posts: 8
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 15 Jul 2017, 19:35
Here was what I did, might be a little faster.

Question is asking
What is remainder of

9^1+9^2....9^8+9^9/6.

Since 9 is 3 more than 6, every multiple of 9 divided 6 will have a remainder of 3.

Therefore 3x9 (for the 9 numbers)= 27/6= 4 with r3.

Answer is 3.
Senior SC Moderator
avatar
V
Joined: 22 May 2016
Posts: 2474
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 16 Jul 2017, 15:47
sunniboy007 wrote:
What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is divided by 6?

A. 0
B. 3
C. 2
D. 5
E. None of the above

I used cyclicity, but I'm not sure it's correct, though it gave me the correct answer.

\(9^1\) = 9
\(9^2\) = 81
\(9^3\) = ..9
\(9^4\)= ...1

Odd powers of nine have a units digit of 9, even powers of nine have a units digit of 1.

There are 5 odd powers of nine (1,3,5,7,9). Units digit is 9, 9*5 = 45, so odd powers' sum will have units digit of 5.

There are four even powers of nine(2,4,6,8). Each ends in 1. 4*1 = 4. Even powers' sum will have a units digit of 4.

Add the two groups: 5 + 4 = 9.

9/6 = remainder 3.

Answer B

You could also pair four odd powers that end in 9 with four even powers that end in 1: 9,1,9,1,9,1,9,1. Each pair sums to 9 + 1 = 10, with a last digit of 0.

Add one more odd power of 9, which will end in 9 (we have five odd powers, we've only used four). 0 + 9 = 9. 9/6 has R3.

I took 33 seconds to answer. I'm a little nervous.

Bunuel - are these methods correct?
_________________

To live is the rarest thing in the world.
Most people just exist.

Oscar Wilde

Director
Director
User avatar
S
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 542
Location: India
Premium Member
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is  [#permalink]

Show Tags

New post 30 Jan 2019, 04:22
Solution:

Given: \(9^1+ 9^2+9^3+ …………………………….+9^9\)

Approach: Let’s take \(N = 9^1+ 9^2+9^3+ …………………………….+9^9\)

We have to find the remainder when N is divided by 6.

\(N = 9^1+ (9^2+9^3+ 9^4+ 9^5+ 9^6+ 9^7+ 9^8+9^9)\)
If we look at the terms in the brackets we have the sum of 8 odd numbers which are multiples of 3. We have to note here that sum of 8 ODD numbers will result in EVEN. [ODD + ODD = EVEN].
So, the sum in brackets is multiple of 6 and obviously, the remainder will be ZERO.
So, we are left with the first term 9, the remainder will be “3” when divided by 6.

The correct answer option is “B”.

_________________

GMAT Mentors
Image

GMAT Club Bot
Re: What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is   [#permalink] 30 Jan 2019, 04:22
Display posts from previous: Sort by

What is the remainder when 9^1 + 9^2 + 9^3 +...+ 9^9 is

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


cron
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®.