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

It is currently 11 Dec 2018, 19:29

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 December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
  • Free GMAT Prep Hour

     December 11, 2018

     December 11, 2018

     09:00 PM EST

     10:00 PM EST

    Strategies and techniques for approaching featured GMAT topics. December 11 at 9 PM EST.
  • The winning strategy for 700+ on the GMAT

     December 13, 2018

     December 13, 2018

     08:00 AM PST

     09:00 AM PST

    What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.

1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?

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

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51100
1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 22 Oct 2014, 23:43
1
12
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

45% (02:21) correct 55% (01:56) wrong based on 299 sessions

HideShow timer Statistics

Senior Manager
Senior Manager
User avatar
Joined: 13 Jun 2013
Posts: 275
Premium Member
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 23 Oct 2014, 00:57
1
1
Bunuel wrote:

Tough and Tricky questions: Remainders.



1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


a number is divisible by 5, if its last digit is divisible by 5

let's look into the sum of last digits of each term of the given expression

1^1=1
2^2=4
3^3=7
4^4=6
5^5=5
6^6=6
7^7=3
8^8=6
9^9=9
10^10=0

adding all these numbers we get 47 which gives a remainder of 2 when divided by 5. so answer must be 2.

bunuel, can you please confirm the answer of this question.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51100
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 23 Oct 2014, 01:06
manpreetsingh86 wrote:
Bunuel wrote:

Tough and Tricky questions: Remainders.



1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


a number is divisible by 5, if its last digit is divisible by 5

let's look into the sum of last digits of each term of the given expression

1^1=1
2^2=4
3^3=7
4^4=6
5^5=5
6^6=6
7^7=3
8^8=6
9^9=9
10^10=0

adding all these numbers we get 47 which gives a remainder of 2 when divided by 5. so answer must be 2.

bunuel, can you please confirm the answer of this question.


Yes, the OA is C. Clicked the wrong button when posting. Edited. Thank you.
_________________

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
Joined: 03 Dec 2015
Posts: 14
1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 16 Dec 2015, 05:28
1
It's not necessary to count 10^10, 5^5, and 1^1 2^2 (1 + 4 is divisible by 5). Realize this thing can save time !
3^3, 4^4, 6^6, 7^7, 8^8, 9^9 plus the last units of them together and get the answer.
Current Student
User avatar
Joined: 15 Mar 2016
Posts: 96
Location: India
Concentration: Operations
GMAT 1: 680 Q47 V36
WE: Engineering (Other)
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 05 May 2016, 20:24
Bunuel wrote:
manpreetsingh86 wrote:
Bunuel wrote:

Tough and Tricky questions: Remainders.



1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


a number is divisible by 5, if its last digit is divisible by 5

let's look into the sum of last digits of each term of the given expression

1^1=1
2^2=4
3^3=7
4^4=6
5^5=5
6^6=6
7^7=3
8^8=6
9^9=9
10^10=0

adding all these numbers we get 47 which gives a remainder of 2 when divided by 5. so answer must be 2.

bunuel, can you please confirm the answer of this question.


Yes, the OA is C. Clicked the wrong button when posting. Edited. Thank you.


I have also done the same way. Although i also tried out by calculating individual remainders
i.e 1=1
4=4
27=2
and so on.

And then I am adding these remainders and then dividing by 5. I am getting the correct ans. But is this approach correct?
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7101
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 05 May 2016, 20:35
tallyho_88 wrote:

I have also done the same way. Although i also tried out by calculating individual remainders
i.e 1=1
4=4
27=2
and so on.

And then I am adding these remainders and then dividing by 5. I am getting the correct ans. But is this approach correct?


Hi,
if the terms are adding as in this Q, you can add up all individual remainders as correctly done by you..
But may be calculating units digit saves time..
say 7^7.. i don't have to get in finding what is 7^7.. I know 7 gives a pattern 7,9,3,1.. so 7th will give same as 4+3rd.. and 3rd is 3 so remainder = 3..
_________________

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

Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8659
Location: Pune, India
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 05 May 2016, 20:54
tallyho_88 wrote:
I have also done the same way. Although i also tried out by calculating individual remainders
i.e 1=1
4=4
27=2
and so on.

And then I am adding these remainders and then dividing by 5. I am getting the correct ans. But is this approach correct?


Yes, you can always do that but division by 2, 5 or 10 is special. All you need is the last digit in these cases. Here is why it is so: http://www.veritasprep.com/blog/2015/12 ... questions/
http://www.veritasprep.com/blog/2015/12 ... ns-part-2/

Using your method, you would have spend a considerable amount of time. Using cylicity would be faster.
_________________

[b]Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

Senior Manager
Senior Manager
avatar
B
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 01 Dec 2016, 04:24
1
1
Bunuel wrote:

Tough and Tricky questions: Remainders.



1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


Different approach with application of modular arithmetic.

\(2^2 = 4 = -1 (mod_5)\)
\(3^3 = 2 (mod_5)\)
\(6 = 1 (mod_5)\)
\(7 = 2 (mod_5)\)
\(8 = 3 (mod_5)\)
\(9 = -1 (mod_5)\)
\(5, 10 = 0 (mod_5)\)

\(\frac{1 - 1 + 2 + (-1)^4 + 0 + 1^6 + 2^7 + 3^8 + (-1)^9 + 0}{5}\)

\(3^2 = (-1) (mod_5)\)

\(\frac{1 - 1 + 2 + 1 + 1 + 1 + (2^2)^3*2 + (3^2)^4 -1}{5}\)

\(= \frac{4 – 2 + 1 – 1}{5} = \frac{2}{5}\)

Remainder 2
Board of Directors
User avatar
P
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4273
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member CAT Tests
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 01 Dec 2016, 08:26
vitaliyGMAT wrote:
Different approach with application of modular arithmetic.

\(2^2 = 4 = -1 (mod_5)\)
\(3^3 = 2 (mod_5)\)
\(6 = 1 (mod_5)\)
\(7 = 2 (mod_5)\)
\(8 = 3 (mod_5)\)
\(9 = -1 (mod_5)\)
\(5, 10 = 0 (mod_5)\)

\(\frac{1 - 1 + 2 + (-1)^4 + 0 + 1^6 + 2^7 + 3^8 + (-1)^9 + 0}{5}\)

\(3^2 = (-1) (mod_5)\)

\(\frac{1 - 1 + 2 + 1 + 1 + 1 + (2^2)^3*2 + (3^2)^4 -1}{5}\)

\(= \frac{4 – 2 + 1 – 1}{5} = \frac{2}{5}\)

Remainder 2

Modular arithmetic is definitely not within the scope of GMAT QA , however knowledge of MODULO ARITHMATIC really helps !!

Kudos for introducing the discussion...

_________________

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: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 01 Dec 2016, 09:06
Abhishek009 wrote:
vitaliyGMAT wrote:
Different approach with application of modular arithmetic.

\(2^2 = 4 = -1 (mod_5)\)
\(3^3 = 2 (mod_5)\)
\(6 = 1 (mod_5)\)
\(7 = 2 (mod_5)\)
\(8 = 3 (mod_5)\)
\(9 = -1 (mod_5)\)
\(5, 10 = 0 (mod_5)\)

\(\frac{1 - 1 + 2 + (-1)^4 + 0 + 1^6 + 2^7 + 3^8 + (-1)^9 + 0}{5}\)

\(3^2 = (-1) (mod_5)\)

\(\frac{1 - 1 + 2 + 1 + 1 + 1 + (2^2)^3*2 + (3^2)^4 -1}{5}\)

\(= \frac{4 – 2 + 1 – 1}{5} = \frac{2}{5}\)

Remainder 2

Modular arithmetic is definitely not within the scope of GMAT QA , however knowledge of MODULO ARITHMATIC really helps !!

Kudos for introducing the discussion...


I agree with you. For many questions, the concept of cyclicity is working pretty good and it’s more than enough. However, GMAT has some questions, which require finding remainders for numbers with composite powers. If we apply cyclicity in this case we’ll need to identify cycles for base as well as for composite power. This is very time consuming and arithmetic error prone approach, taking into consideration time pressure. In this case, modulo arithmetic and theorems of number theory can significantly simplify our lives.
Thanks for kudos buddy :-D
Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 9120
Premium Member
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?  [#permalink]

Show Tags

New post 19 Aug 2018, 03:31
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

GMAT Club Bot
Re: 1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder? &nbs [#permalink] 19 Aug 2018, 03:31
Display posts from previous: Sort by

1^1 + 2^2 + 3^3 + ... + 10^10 is divided by 5. What is the remainder?

  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®.