It is currently 19 Nov 2017, 09:24

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

If n is a positive integer, what is the remainder when

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

Hide Tags

3 KUDOS received
Manager
Manager
avatar
Joined: 07 Oct 2005
Posts: 141

Kudos [?]: 147 [3], given: 0

Location: Boston,MA
If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 27 Nov 2007, 14:38
3
This post received
KUDOS
14
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

64% (01:09) correct 36% (01:42) wrong based on 634 sessions

HideShow timer Statistics

If n is a positive integer, what is the remainder when (7^(4n+3))(6^n) is divided by 10?
A. 1
B. 2
C. 4
D. 6
E. 8
[Reveal] Spoiler: OA

_________________

--gregspirited

Kudos [?]: 147 [3], given: 0

Expert Post
6 KUDOS received
CEO
CEO
User avatar
Joined: 17 Nov 2007
Posts: 3583

Kudos [?]: 4662 [6], given: 360

Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
GMAT ToolKit User Premium Member
 [#permalink]

Show Tags

New post 27 Nov 2007, 14:57
6
This post received
KUDOS
Expert's post
2
This post was
BOOKMARKED
E (J). 8

last digits of 7^m
7^1 : 7
7^2 : 9
7^3 : 3
7^4 : 1
7^5 : 7
period=4 ==> last digit: 7^(4n+3) = 7^(4+3) = 3
last digit of 6^n always 6.

3*6 ==> 8 - reminder (last digit)

Kudos [?]: 4662 [6], given: 360

CEO
CEO
User avatar
Joined: 29 Mar 2007
Posts: 2553

Kudos [?]: 528 [0], given: 0

Re: PS - remaineder of (7^(4n+3))(6^n) is divided by 10 [#permalink]

Show Tags

New post 27 Nov 2007, 15:37
gregspirited wrote:
If n is a positive integer, what is the remainder
When (7^(4n+3))(6^n) is divided by 10?
F. 1
G. 2
H. 4
I. 6
J. 8


This one took me bout 3 1/2 min. Just testin numbers and what not.

First notice that n is positive. Save time by noticing that :P I worked out one solution where n=0 only to find that thats not an option :p.
1-7 stands for ^1 thru 7
1: 7*1=7
2: 7*7=9
3: 7*9=3
4: 7*3=1
5: 7*1=7
6: 7*7=9
7: 7*9=3

Pattern repeats every @ 5. Notice every ^4 or multiple of 4 is always going to be 1. This is just for future notice for similar problems.

so 7^4n+3 ---> if n=1 then its ((7^7)*6))/10 which can say is going to be 3*8--> 18/10 --> R=8

Now from here if id double check just to make sure.

7^4(2)+3*6^2 ---> 7^11*36 or we can just say again 7^11*6 (b/c we are only interested in the units digit).

Since ^12 is going to be 1 that means ^11 is going to be 3 (as taken from our pattern)

so again 3*6=18/10 ---> R =8.


E or J in this problem.

Kudos [?]: 528 [0], given: 0

2 KUDOS received
Director
Director
avatar
Joined: 09 Aug 2006
Posts: 754

Kudos [?]: 260 [2], given: 0

Re: PS - remaineder of (7^(4n+3))(6^n) is divided by 10 [#permalink]

Show Tags

New post 27 Nov 2007, 22:57
2
This post received
KUDOS
gregspirited wrote:
If n is a positive integer, what is the remainder
When (7^(4n+3))(6^n) is divided by 10?
F. 1
G. 2
H. 4
I. 6
J. 8


J

All we need to do to find the remainder is find out the units digit of the expression.

7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1

If n = 1 then 7^(4n+3) = 7^7 = 7^4*7^3
units digit of 7^4 which is 1 * units digits of 7^3 which is 3 = 3
6 raised to any non 0 positive power will have units digit of 6
therefore units digit of expression = 6*3 = 8
when divided by 10 this will always leave a remainder of 8.

Kudos [?]: 260 [2], given: 0

Director
Director
avatar
Joined: 11 Jun 2007
Posts: 909

Kudos [?]: 291 [0], given: 0

Re: PS - remaineder of (7^(4n+3))(6^n) is divided by 10 [#permalink]

Show Tags

New post 27 Nov 2007, 23:15
gregspirited wrote:
If n is a positive integer, what is the remainder
When (7^(4n+3))(6^n) is divided by 10?
F. 1
G. 2
H. 4
I. 6
J. 8


i got 8 after working it out in less than a minute

7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
repeats we're looking for 3rd: 3
6^n = always ends in 6
3*6 = ends in 8

when divided by 10 will leave a remainder of 8

Kudos [?]: 291 [0], given: 0

Manager
Manager
User avatar
Joined: 06 Feb 2010
Posts: 168

Kudos [?]: 1434 [0], given: 182

Concentration: Marketing, Leadership
Schools: University of Dhaka - Class of 2010
GPA: 3.63
WE: Business Development (Consumer Products)
If n is a positive integer, what is the remainder when ((7^( [#permalink]

Show Tags

New post 14 Nov 2010, 23:40
If n is a positive integer, what is the remainder when ((7^(4n+3)(6^n)) is divided by 10?

(A) 1
(B) 2
(C) 4
(D) 6
(E) 8
_________________

Practice Makes a Man Perfect. Practice. Practice. Practice......Perfectly

Critical Reasoning: http://gmatclub.com/forum/best-critical-reasoning-shortcuts-notes-tips-91280.html

Collections of MGMAT CAT: http://gmatclub.com/forum/collections-of-mgmat-cat-math-152750.html

MGMAT SC SUMMARY: http://gmatclub.com/forum/mgmat-sc-summary-of-fourth-edition-152753.html

Sentence Correction: http://gmatclub.com/forum/sentence-correction-strategies-and-notes-91218.html

Arithmatic & Algebra: http://gmatclub.com/forum/arithmatic-algebra-93678.html

Helpful Geometry formula sheet: http://gmatclub.com/forum/best-geometry-93676.html


I hope these will help to understand the basic concepts & strategies. Please Click ON KUDOS Button.

Kudos [?]: 1434 [0], given: 182

Manager
Manager
avatar
Joined: 02 Apr 2010
Posts: 102

Kudos [?]: 141 [0], given: 18

Re: remainder when ((7^(4n+3)(6^n)) [#permalink]

Show Tags

New post 15 Nov 2010, 00:27
Use n = 1 and plug into the equation => 7^7*6^1 / 10

Check the cyclicity of n:
7^1 = 7
7^2 = 49
7^3 = ...9*7 =...63
7^4 = ...3*7 =...21
7^5 = ...1*7 =....7 (last digit the same as for 7^1)
=> 7^7 = ....3

....3*6^1 = ....18. Since the number ends in 8 the remainder when divided by 10 must be 8.

Hence, solution E is correct.

Kudos [?]: 141 [0], given: 18

Director
Director
User avatar
Status: No dream is too large, no dreamer is too small
Joined: 14 Jul 2010
Posts: 609

Kudos [?]: 1154 [0], given: 39

Re: If n is a positive integer, what is the remainder When [#permalink]

Show Tags

New post 18 Jan 2012, 12:54
Cyclicity of 7 is 4.
if n = 1, power of 7 will 7 and, the reminder will 3 if 7 is divided by 4
if n = 3, power of 7 will 15 and, the reminder will 3 if 15 is divided by 4
Last digit of 7^4 will 3, and last digit 6 power anything will 6
Now, 6*3 = 18/10 = 8 Reminder.
Ans. E
_________________

Collections:-
PSof OG solved by GC members: http://gmatclub.com/forum/collection-ps-with-solution-from-gmatclub-110005.html
DS of OG solved by GC members: http://gmatclub.com/forum/collection-ds-with-solution-from-gmatclub-110004.html
100 GMAT PREP Quantitative collection http://gmatclub.com/forum/gmat-prep-problem-collections-114358.html
Collections of work/rate problems with solutions http://gmatclub.com/forum/collections-of-work-rate-problem-with-solutions-118919.html
Mixture problems in a file with best solutions: http://gmatclub.com/forum/mixture-problems-with-best-and-easy-solutions-all-together-124644.html

Kudos [?]: 1154 [0], given: 39

Director
Director
avatar
Joined: 29 Nov 2012
Posts: 866

Kudos [?]: 1451 [0], given: 543

Re: remainder when ((7^(4n+3)(6^n)) [#permalink]

Show Tags

New post 10 Aug 2013, 00:44
6 and 5 are 2 numbers that have the same units digit all the time \(6^6 = 6\) \(5^5 = 5\)

So in this question 6^n = 6

\(7^{4n+3}\) test some values say n=0,1 etc

we get a pattern if 0 then it becomes\(7^3\)

if n = 1 then it becomes \(7^7\)


The pattern for 7 is {7,9,3,1}

Units digit 7^{4n+3} = 3 and units digit of 6^n = 6

so 18 divided by 10 remainder is 8

Answer E
_________________

Click +1 Kudos if my post helped...

Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/

GMAT Prep software What if scenarios http://gmatclub.com/forum/gmat-prep-software-analysis-and-what-if-scenarios-146146.html

Kudos [?]: 1451 [0], given: 543

Senior Manager
Senior Manager
avatar
Joined: 10 Jul 2013
Posts: 326

Kudos [?]: 426 [0], given: 102

Re: remainder when ((7^(4n+3)(6^n)) [#permalink]

Show Tags

New post 10 Aug 2013, 02:39
monirjewel wrote:
If n is a positive integer, what is the remainder when ((7^(4n+3)(6^n)) is divided by 10?
(A) 1
(B) 2
(C) 4
(D) 6
(E) 8

..................
7^4n × 7^3 × 6^n
= (7^2)2n × 343 × 6^n
=(50-1)^2n × 343 × 6^n
so the last term = (-1)^2n × 343 × 6^n = 343 × 6^n
For any values of n 6^n = something 6 in the unit digit, and 343 × something 6 in the unit digit will always provide something 8 in the unit digit,
so Answer is E
_________________

Asif vai.....

Kudos [?]: 426 [0], given: 102

Intern
Intern
avatar
Joined: 05 Mar 2014
Posts: 9

Kudos [?]: [0], given: 275

Schools: Ross '18
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 06 Apr 2014, 07:15
Answer is E = 8

((7^(4n+3))*6^n)/10

Trying for 1 we have:

((7^7)*(6^1))/10 = (7*6*(7^6))/10 = (42*(7^6))/10 = 4.2*(7^6).

7^2 = 9
7^3 = 9*7 = 3
7^4 = 3*7 = 1
7^5 = 1*7 = 7
7^6 = 7*7 = 9

9*2"from 4.2" gives us 8

Kudos [?]: [0], given: 275

1 KUDOS received
Verbal Forum Moderator
User avatar
Joined: 10 Oct 2012
Posts: 627

Kudos [?]: 1387 [1], given: 136

Premium Member
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 06 Apr 2014, 07:38
1
This post received
KUDOS
gregspirited wrote:
If n is a positive integer, what is the remainder when (7^(4n+3))(6^n) is divided by 10?
A. 1
B. 2
C. 4
D. 6
E. 8


Theory : The cyclicity for the unit's digit for 7 repeats at an interval of 4. Thus, units digit for \(7^1 = 7, 7^2 = 9 , 7^3 = 3\) and \(7^4 = 1\)

Given expression : \(7^{4n}*7^{3}*6^n\) and note that n is a positive integer.

As 4n is always a multiple of 4, the units digit of \(7^{4n}\) will always be 1. Units digit of\(7^3 = 3\). Also, \(6^n\) will always have the same units digit of 6, just as\(5^n\)(units digit of 5) does.

Thus, final expression will have the unit's digit as : \(1*3*6 = 18\). As the divisor is 10, the remainder will always be the units digit = 8.

E.
_________________

All that is equal and not-Deep Dive In-equality

Hit and Trial for Integral Solutions

Kudos [?]: 1387 [1], given: 136

SVP
SVP
User avatar
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1852

Kudos [?]: 2712 [0], given: 193

Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 07 Apr 2014, 19:55
\(7^{4n+3} * 6^n\)

Placing value of n=1

\(= 7^7 * 6\)

\(= 7^7 (10 - 4)\)

\(= 7^7 * 10 - 7^7 * 4\)

\(7^7 * 10\) >> Will not leave any remainder

\(- 7^7 * 4\) >> Cyclicity for power 7 = 7, 9, 3, 1

\(7^7\) gives 3 in the units place & multiplying by 4 gives 2 in units place

So 10 - 2 = 8

Answer = E
_________________

Kindly press "+1 Kudos" to appreciate :)

Kudos [?]: 2712 [0], given: 193

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15658

Kudos [?]: 282 [0], given: 0

Premium Member
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 08 Apr 2015, 10:52
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 [?]: 282 [0], given: 0

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15658

Kudos [?]: 282 [0], given: 0

Premium Member
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 16 Apr 2016, 14:31
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

Kudos [?]: 282 [0], given: 0

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15658

Kudos [?]: 282 [0], given: 0

Premium Member
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 08 Sep 2017, 02:17
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 [?]: 282 [0], given: 0

Expert Post
Target Test Prep Representative
User avatar
S
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 1806

Kudos [?]: 922 [0], given: 3

Location: United States (CA)
Re: If n is a positive integer, what is the remainder when [#permalink]

Show Tags

New post 11 Sep 2017, 16:01
gregspirited wrote:
If n is a positive integer, what is the remainder when (7^(4n+3))(6^n) is divided by 10?
A. 1
B. 2
C. 4
D. 6
E. 8


Simplifying the expression, we have:

(7^4n)(7^3)(6^n)

We need to determine the remainder when the expression above is divided by 10; the remainder will be equal to the units digit of that expression.

Since 6^n will always have a units digit of 6, let’s determine the units digits of (7^4n) and (7^3):

Let’s examine the units digits of 7^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are concerned ONLY with the units digit of 7 raised to a power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

The pattern of the units digits of powers of 7 repeats every 4 exponents. The pattern is 7–9–3–1. In this pattern, all positive exponents that are multiples of 4 will produce 1 as its units digit. Thus:

7^3 has a units digit of 3 and 7^4n has a units digit of 1.

Since 1 x 3 x 6 = 18, (7^4n)(7^3)(6^n) has a units digit of 8, which is also the remainder when (7^4n)(7^3)(6^n) is divided by 10.

Answer: E
_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Kudos [?]: 922 [0], given: 3

Re: If n is a positive integer, what is the remainder when   [#permalink] 11 Sep 2017, 16:01
Display posts from previous: Sort by

If n is a positive integer, what is the remainder when

  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®.