It is currently 20 Nov 2017, 01:00

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 unit's digit of 7^75 + 6 ?

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

Hide Tags

Manager
Manager
avatar
Joined: 22 Jun 2010
Posts: 56

Kudos [?]: 78 [0], given: 10

What is the unit's digit of 7^75 + 6 ? [#permalink]

Show Tags

New post 14 Sep 2010, 12:58
8
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  15% (low)

Question Stats:

67% (00:37) correct 33% (00:48) wrong based on 729 sessions

HideShow timer Statistics

What is the unit's digit of \(7^{75} + 6\) ?

A. 1
B. 3
C. 5
D. 7
E. 9

(C) 2008 GMAT Club - m12#29

I put the official explanation and the part I do not understand (blue text) in a spoiler

[Reveal] Spoiler:
\(7^1\) ends with 7

\(7^2\) ends with 9

\(7^3\) ends with 3

\(7^4\) ends with 1

\(7^5\) ends with 7

...

\(7^{76}\) ends with 1. --> ???

So, \(7^{75}\) ends with 3. --> ???
If 7^5 ends with 7, shouldnt 7^75 also end with 7? Hence 7+6=13 - answer b?? Please help!!


\(7^{75} + 6\) ends with 9.
The correct answer is E.
[Reveal] Spoiler: OA

Last edited by Bunuel on 03 Jul 2013, 00:52, edited 1 time in total.
Renamed the topic and edited the question.

Kudos [?]: 78 [0], given: 10

Expert Post
3 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42259

Kudos [?]: 132744 [3], given: 12370

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

Show Tags

New post 14 Sep 2010, 13:20
3
This post received
KUDOS
Expert's post
AndreG wrote:
What is the unit's digit of \(7^{75} + 6\) ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

[Reveal] Spoiler:
\(7^1\) ends with 7

\(7^2\) ends with 9

\(7^3\) ends with 3

\(7^4\) ends with 1

\(7^5\) ends with 7

...

\(7^{76}\) ends with 1. --> ???

So, \(7^{75}\) ends with 3. --> ???
If 7^5 ends with 7, shouldnt 7^75 also end with 7? Hence 7+6=13 - answer b?? Please help!!


\(7^{75} + 6\) ends with 9.
The correct answer is E.


7 in power repeats pattern of 4: 7-9-3-1. As 75=4*18+3 then the last digit of \(7^{75}\) is the same as the last digit of \(7^3\), which is 3. Units digit of \(7^{75} + 6\) will be: 3 plus 6 = 9.

Answer: E.

For more on this issue check Number Theory chapter of Math Book (link in my signature).

Hope it helps.
_________________

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

Kudos [?]: 132744 [3], given: 12370

Senior Manager
Senior Manager
avatar
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 292

Kudos [?]: 57 [0], given: 7

Schools: Chicago,Tuck,Oxford,cambridge
Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

Show Tags

New post 14 Sep 2010, 13:26
AndreG wrote:
What is the unit's digit of \(7^{75} + 6\) ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

[Reveal] Spoiler:
\(7^1\) ends with 7

\(7^2\) ends with 9

\(7^3\) ends with 3

\(7^4\) ends with 1

\(7^5\) ends with 7

...

\(7^{76}\) ends with 1. --> ???

So, \(7^{75}\) ends with 3. --> ???
If 7^5 ends with 7, shouldnt 7^75 also end with 7? Hence 7+6=13 - answer b?? Please help!!


\(7^{75} + 6\) ends with 9.
The correct answer is E.



well i will say that whatever may be the number if we have to find the last digit of some number whose power isgiven..then the best method is to divide the power by 4 since all the digits from 1...9 will surely repeat after every 4th digit...
then raise the digit to the power of remainder...
here 75/4 remainder=3
7^3=last digit comes out to be 3
now 3+6=9

thanx

Kudos [?]: 57 [0], given: 7

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42259

Kudos [?]: 132744 [0], given: 12370

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

Show Tags

New post 14 Sep 2010, 13:42
Expert's post
4
This post was
BOOKMARKED
sandeep800 wrote:
AndreG wrote:
What is the unit's digit of \(7^{75} + 6\) ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

[Reveal] Spoiler:
\(7^1\) ends with 7

\(7^2\) ends with 9

\(7^3\) ends with 3

\(7^4\) ends with 1

\(7^5\) ends with 7

...

\(7^{76}\) ends with 1. --> ???

So, \(7^{75}\) ends with 3. --> ???
If 7^5 ends with 7, shouldnt 7^75 also end with 7? Hence 7+6=13 - answer b?? Please help!!


\(7^{75} + 6\) ends with 9.
The correct answer is E.



well i will say that whatever may be the number if we have to find the last digit of some number whose power isgiven..then the best method is to divide the power by 4 since all the digits from 1...9 will surely repeat after every 4th digit...
then raise the digit to the power of remainder...
here 75/4 remainder=3
7^3=last digit comes out to be 3
now 3+6=9

thanx


The above is correct with a little correction: when remainder is zero, then we should rise to the power not of remainder 0 but to the power of the cyclicity number.

For example las digit of 7^24 is the same as the last digit of 7^4 as the cyclicity of 7 in power is 4 and 24 divided by 4 gives remainder of zero.

From Number Theory chapter of Math Book:

LAST DIGIT OF A POWER

Determining the last digit of \((xyz)^n\):

1. Last digit of \((xyz)^n\) is the same as that of \(z^n\);
2. Determine the cyclicity number \(c\) of \(z\);
3. Find the remainder \(r\) when \(n\) divided by the cyclisity;
4. When \(r>0\), then last digit of \((xyz)^n\) is the same as that of \(z^r\) and when \(r=0\), then last digit of \((xyz)^n\) is the same as that of \(z^c\), where \(c\) is the cyclisity number.

• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.
• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.
• Integers ending with 4 (eg. \((xy4)^n\)) have a cyclisity of 2. When n is odd \((xy4)^n\) will end with 4 and when n is even \((xy4)^n\) will end with 6.
• Integers ending with 9 (eg. \((xy9)^n\)) have a cyclisity of 2. When n is odd \((xy9)^n\) will end with 9 and when n is even \((xy9)^n\) will end with 1.

Example: What is the last digit of \(127^{39}\)?
Solution: Last digit of \(127^{39}\) is the same as that of \(7^{39}\). Now we should determine the cyclisity of \(7\):

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of \(127^{39}\) is the same as that of the last digit of \(7^{39}\), is the same as that of the last digit of \(7^3\), which is \(3\).

Hope it helps.
_________________

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

Kudos [?]: 132744 [0], given: 12370

Senior Manager
Senior Manager
avatar
Status: GMAT Time...!!!
Joined: 03 Apr 2010
Posts: 292

Kudos [?]: 57 [0], given: 7

Schools: Chicago,Tuck,Oxford,cambridge
Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

Show Tags

New post 14 Sep 2010, 13:55
1
This post was
BOOKMARKED
Bunuel wrote:
sandeep800 wrote:
AndreG wrote:
What is the unit's digit of \(7^{75} + 6\) ?

(C) 2008 GMAT Club - m12#29

* 1
* 3
* 5
* 7
* 9

I put the official explanation and the part I do not understand (blue text) in a spoiler

[Reveal] Spoiler:
\(7^1\) ends with 7

\(7^2\) ends with 9

\(7^3\) ends with 3

\(7^4\) ends with 1

\(7^5\) ends with 7

...

\(7^{76}\) ends with 1. --> ???

So, \(7^{75}\) ends with 3. --> ???
If 7^5 ends with 7, shouldnt 7^75 also end with 7? Hence 7+6=13 - answer b?? Please help!!


\(7^{75} + 6\) ends with 9.
The correct answer is E.



well i will say that whatever may be the number if we have to find the last digit of some number whose power isgiven..then the best method is to divide the power by 4 since all the digits from 1...9 will surely repeat after every 4th digit...
then raise the digit to the power of remainder...
here 75/4 remainder=3
7^3=last digit comes out to be 3
now 3+6=9

thanx


The above is correct with a little correction: when remainder is zero, then we should rise to the power not of remainder 0 but to the power of the cyclicity number.

For example las digit of 7^24 is the same as the last digit of 7^4 as the cyclicity of 7 in power is 4 and 24 divided by 4 gives remainder of zero.

From Number Theory chapter of Math Book:

LAST DIGIT OF A POWER

Determining the last digit of \((xyz)^n\):

1. Last digit of \((xyz)^n\) is the same as that of \(z^n\);
2. Determine the cyclicity number \(c\) of \(z\);
3. Find the remainder \(r\) when \(n\) divided by the cyclisity;
4. When \(r>0\), then last digit of \((xyz)^n\) is the same as that of \(z^r\) and when \(r=0\), then last digit of \((xyz)^n\) is the same as that of \(z^c\), where \(c\) is the cyclisity number.

• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.
• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.
• Integers ending with 4 (eg. \((xy4)^n\)) have a cyclisity of 2. When n is odd \((xy4)^n\) will end with 4 and when n is even \((xy4)^n\) will end with 6.
• Integers ending with 9 (eg. \((xy9)^n\)) have a cyclisity of 2. When n is odd \((xy9)^n\) will end with 9 and when n is even \((xy9)^n\) will end with 1.

Example: What is the last digit of \(127^{39}\)?
Solution: Last digit of \(127^{39}\) is the same as that of \(7^{39}\). Now we should determine the cyclisity of \(7\):

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of \(127^{39}\) is the same as that of the last digit of \(7^{39}\), is the same as that of the last digit of \(7^3\), which is \(3\).

Hope it helps.


Thanx a lot bunuel for correcting me..i wud have applied my method in GMAT if u had not corrected me....:)

Kudos [?]: 57 [0], given: 7

Manager
Manager
avatar
Joined: 22 Jun 2010
Posts: 56

Kudos [?]: 78 [0], given: 10

Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

Show Tags

New post 14 Sep 2010, 14:06
Wow, you guys helped me a lot! THANKS!!

Kudos [?]: 78 [0], given: 10

Senior Manager
Senior Manager
avatar
Joined: 20 Jul 2010
Posts: 256

Kudos [?]: 101 [0], given: 9

GMAT ToolKit User Reviews Badge
Re: GMAT CLUB TEST m12#29 - last digit [#permalink]

Show Tags

New post 14 Sep 2010, 15:23
Thanks for summarising the concept. I used to calculate what you call cyclicity in every problem and reach my conclusions
_________________

If you like my post, consider giving me some KUDOS !!!!! Like you I need them

Kudos [?]: 101 [0], given: 9

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42259

Kudos [?]: 132744 [0], given: 12370

Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

Show Tags

New post 09 Mar 2014, 13:09

Kudos [?]: 132744 [0], given: 12370

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15638

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

Premium Member
Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

Show Tags

New post 02 Apr 2015, 18:38
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

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15638

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

Premium Member
Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

Show Tags

New post 06 Jun 2016, 18:59
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

Board of Directors
User avatar
G
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 3098

Kudos [?]: 1115 [0], given: 327

Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member
Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

Show Tags

New post 23 Nov 2016, 11:45
AndreG wrote:
What is the unit's digit of \(7^{75} + 6\) ?

A. 1
B. 3
C. 5
D. 7
E. 9


Since , the cyclicity of 7 is 4

The units digit of \(7^{75} = 3\)

So, Units digit will be 3+ 6 = 9

Hence, answer will be (E) 9...


_________________

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 )

Kudos [?]: 1115 [0], given: 327

Retired Moderator
avatar
P
Joined: 12 Aug 2015
Posts: 2213

Kudos [?]: 871 [0], given: 602

GMAT ToolKit User Premium Member
Re: What is the unit's digit of 7^75 + 6 ? [#permalink]

Show Tags

New post 23 Jan 2017, 18:22
Nice Question.
Here is what i did in this one ->
Cyclicity of 7 is 4 =>
7
9
3
1
Hence the units digit of 7^75 => 7^4m+3 will be 3.
So 7^75+6 will have 3+6=9 as its units digit.

Hence E.

_________________

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

Kudos [?]: 871 [0], given: 602

Re: What is the unit's digit of 7^75 + 6 ?   [#permalink] 23 Jan 2017, 18:22
Display posts from previous: Sort by

What is the unit's digit of 7^75 + 6 ?

  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®.