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

It is currently 17 Jan 2019, 19:38

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 January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
  • The winning strategy for a high GRE score

     January 17, 2019

     January 17, 2019

     08:00 AM PST

     09:00 AM PST

    Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL.
  • Free GMAT Strategy Webinar

     January 19, 2019

     January 19, 2019

     07:00 AM PST

     09:00 AM PST

    Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.

What is the units digit of (3^{101})(7^{103})?

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

Hide Tags

 
Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 6811
GMAT 1: 760 Q51 V42
GPA: 3.82
Premium Member
What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 00:56
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

68% (00:54) correct 32% (01:08) wrong based on 158 sessions

HideShow timer Statistics

[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)

_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Manager
Manager
avatar
G
Joined: 14 Jun 2018
Posts: 223
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 01:04
3^101 * 7^103 = 21^101 * 49
cycle of last digit of 21 is 1.
Ans E
Director
Director
User avatar
G
Joined: 20 Feb 2015
Posts: 795
Concentration: Strategy, General Management
Premium Member
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 01:06
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


\((3^{101})(7^{103})\)

101= 4k+1
103=4k+3

above can be written as

\((3^{1})(7^{3})\)
3*343
units digit = 9
Senior Manager
Senior Manager
avatar
S
Joined: 04 Aug 2010
Posts: 320
Schools: Dartmouth College
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 02:38
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.

\(3^{101}\):
3¹ --> units digit of 3.
3² --> units digit of 9. (Since the product of the preceding units digit and 3 = 3*3 = 9.)
3³ --> units digit of 7. (Since the product of the preceding units digit and 3 = 9*3 = 27.)
3⁴ --> units digit of 1. (Since the product of the preceding units digit and 3 = 7*3 = 21.)
From here, the units digits will repeat in the same pattern: 3, 9, 7, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 3 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, \(3^{100}\)has a units digit of 1.
From here, the cycle of units digits will repeat: 3, 9, 7, 1...
Thus, \(3^{101}\) has a units digit of 3.

7¹⁰³:
7¹ --> units digit of 7.
7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
7⁴ --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)
From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, \(7^{100}\) has a units digit of 1.
From here, the cycle of units digits will repeat: 7, 9, 3, 1...
\(7^{101}\)--> units digit of 7.
\(7^{102}\) --> units digit of 9.
\(7^{103}\)--> units digit of 3.

Result:
\(3^{101}7^{103}\) = (integer with a units digit of 3)(integer with a units digit of 3) = integer with a units digit of 9.


_________________

GMAT and GRE Tutor
Over 1800 followers
Click here to learn more
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.

Board of Directors
User avatar
P
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4330
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member CAT Tests
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 06:40
1
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


\(3^4\) = Units digit 1
\(7^4\) = Units digit 1

\((3^{101})(7^{103})\)

= \((3^{4*25}*3^1)(7^{4*25}*7^3)\)

3^1 will have units digit 1 and 7^3 will have units digit 3

So, The units digit of the expression will be \(1*3*1*3 = 9\) , Answer must be (E)
_________________

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 )

VP
VP
User avatar
D
Joined: 09 Mar 2016
Posts: 1287
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 07:48
Abhishek009 wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


\(3^4\) = Units digit 1
\(7^4\) = Units digit 1

\((3^{101})(7^{103})\)

= \((3^{4*25}*3^1)(7^{4*25}*7^3)\)

3^1 will have units digit 1 and 7^3 will have units digit 3

So, The units digit of the expression will be \(1*3*1*3 = 9\) , Answer must be (E)



hey there Abhishek009 :) hope your solo guitar career is thriving :-) let me know when you are gving your next rock concert , i will buy tickets :grin: :lol:

if 3^1 will have units digit 1 and 7^3 will have units digit 3

So we have unit digit 1 and unit digit 3 hence 1*3 = 3 :? ?

where from did you get so many numbers \(1*3*1*3 = 9\) :?


have a great evening :)
Board of Directors
User avatar
P
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4330
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member CAT Tests
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 29 Aug 2018, 08:06
1
dave13 wrote:

hey there Abhishek009 :) hope your solo guitar career is thriving :-) let me know when you are gving your next rock concert , i will buy tickets :grin: :lol:

if 3^1 will have units digit 1 and 7^3 will have units digit 3

So we have unit digit 1 and unit digit 3 hence 1*3 = 3 :? ?

where from did you get so many numbers \(1*3*1*3 = 9\) :?


have a great evening :)


\(= (3^{4∗25}∗3^1)(7^{4∗25}∗7^3)\)

\(= (1^{25}∗3)(1^{25}∗3)\) { Units digit of 3^4 = 1 and Units digit of 7^4 = 3 }

\(= 1* 3 * 1 * 3\)

Hope this helps!!!

PS : Mr Bean is my fav character, and I really love your innocent looking DP, good evening friend, plz feel free to revert in case of any further doubt...
_________________

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 )

Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 6811
GMAT 1: 760 Q51 V42
GPA: 3.82
Premium Member
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 31 Aug 2018, 00:15
=>

The units digit is the remainder when \((3^{101})(7^{103})\) is divided by \(10\).

The remainders when powers of \(3\) are divided by \(10\) are
\(3^1: 3,\)
\(3^2: 9,\)
\(3^3: 7,\)
\(3^4: 1,\)
\(3^5: 3,\)

So, the units digits of \(3^n\) have period \(4\):
They form the cycle \(3 -> 9 -> 7 -> 1.\)
Thus, \(3^n\) has the units digit of \(3\) if \(n\) has a remainder of \(1\) when it is divided by \(4\).
The remainder when \(101\) is divided by \(4\) is \(1\), so the units digit of \(3^{101}\) is \(3\).

The remainders when powers of \(7\) are divided by \(10\) are
\(7^1: 7,\)
\(7^2: 9,\)
\(7^3: 3,\)
\(7^4: 1,\)
\(7^5: 7,\)

So, the units digits of \(7^n\) have period \(4\):
They form the cycle \(7 -> 9 -> 3 -> 1\).
Thus, \(7^n\) has the units digit of \(3\) if \(n\) has a remainder of \(3\) when it is divided by \(4\).
The remainder when \(103\) is divided by \(4\) is \(3\), so the units digit of \(7^{103}\) is \(3\).

Thus, the units digit of \((3^{101})(7^{103})\) is \(3*3 = 9.\)

Therefore, the answer is E.
Answer: E
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Intern
Intern
User avatar
B
Joined: 16 Jul 2017
Posts: 24
Location: India
Concentration: Finance, Economics
What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post Updated on: 03 Oct 2018, 09:34
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)



3^101 * 7^103

cycle of 3 is 3,9,7,1
101st through cyclicity will be 3
cycle of 7 is 7,9,3,1
103rd through cyclicity will be 3

Units digit i.e. 3*3 = 9

Ans (E)

Originally posted by Natty97 on 31 Aug 2018, 09:00.
Last edited by Natty97 on 03 Oct 2018, 09:34, edited 1 time in total.
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2830
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 03 Sep 2018, 18:01
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


Let’s start by evaluating the pattern of the units digits of 3^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 3. When writing out the pattern, notice that we are concerned ONLY with the units digit of 3 raised to each power.

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

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

3^100 has a units digit of 1, and so 3^101 has a units digit of 3.

Next, we can evaluate the pattern of 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 each power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

The pattern of the units digit 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:

7^104 has a units digit of 1, and so 7^103 has a units digit of 3.

Thus, the units digit of 3^101 x 7^103 is 3 x 3 = 9.

Answer: E
_________________

Jeffery Miller
Head of GMAT Instruction

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

CEO
CEO
User avatar
D
Joined: 11 Sep 2015
Posts: 3331
Location: Canada
Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

Show Tags

New post 03 Oct 2018, 07:31
Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of \((3^{101})(7^{103})\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


This is a great question for applying the following property: (x^n)(y^n) = (xy)^n
For example, (3^7)(5^7) = 15^7

(3^101)(7^103) = (3^101)(7^101)(7^2) [rewrote 7^103 as the product of 7^101 and 7^2]
= (3^101)(7^101)(49) [evaluated 7^2]
= (21^101)(49) [applied above property]

Notice that 21^n will have units digit 1 for all positive integer values of n
So, = 21^101 = --------1 [some big number ending in 1]
So, we get:
= (--------1)(49)
= --------9

Answer: E

RELATED VIDEO FROM OUR COURSE

_________________

Test confidently with gmatprepnow.com
Image

GMAT Club Bot
Re: What is the units digit of (3^{101})(7^{103})? &nbs [#permalink] 03 Oct 2018, 07:31
Display posts from previous: Sort by

What is the units digit of (3^{101})(7^{103})?

  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®.