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

It is currently 15 Dec 2018, 14:59

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
  • $450 Tuition Credit & Official CAT Packs FREE

     December 15, 2018

     December 15, 2018

     10:00 PM PST

     11:00 PM PST

    Get the complete Official GMAT Exam Pack collection worth $100 with the 3 Month Pack ($299)
  • FREE Quant Workshop by e-GMAT!

     December 16, 2018

     December 16, 2018

     07:00 AM PST

     09:00 AM PST

    Get personalized insights on how to achieve your Target Quant Score.

If n = (33)^43 + (43)^33 what is the units digit of n?

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

Hide Tags

Director
Director
avatar
Status: Final Lap Up!!!
Affiliations: NYK Line
Joined: 21 Sep 2012
Posts: 946
Location: India
GMAT 1: 410 Q35 V11
GMAT 2: 530 Q44 V20
GMAT 3: 630 Q45 V31
GPA: 3.84
WE: Engineering (Transportation)
If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

Show Tags

New post Updated on: 04 Oct 2012, 03:23
12
73
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

61% (00:50) correct 39% (00:54) wrong based on 3054 sessions

HideShow timer Statistics

If n = (33)^43 + (43)^33 what is the units digit of n?

A. 0
B. 2
C. 4
D. 6
E. 8

Originally posted by Archit143 on 03 Oct 2012, 23:13.
Last edited by Bunuel on 04 Oct 2012, 03:23, edited 1 time in total.
Moved to PS forum.
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51218
Re: If n = (33)^43  [#permalink]

Show Tags

New post 09 Mar 2014, 12:04
8
36
Bunuel wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?

A. 0
B. 2
C. 4
D. 6
E. 8

First of all, the units digit of (33)^43 is the same as that of 3^43 and the units digit of (43)^33 is the same as that of 3^33. So, we need to find the units digit of 3^43 + 3^33.

Next, the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}:
3^1=3 (the units digit is 3)
3^2=9 (the units digit is 9)
3^3=27 (the units digit is 7)
3^4=81 (the units digit is 1)
3^5=243 (the units digit is 3 again!)
...

Thus:
The units digit of 3^43 is the same as the units digit of 3^3, so 7 (43 divided by the cyclicity of 4 gives the remainder of 3).
The units digit of 3^33 is the same as the units digit of 3^1, so 3 (33 divided by the cyclicity of 4 gives the remainder of 1).

Therefore the units digit of (33)^43 + (43)^33 is 7 + 3 = 0.

Answer: A.


For more on this check theory and problems listed below:

Problem Solving:


650+


what-is-the-tens-digit-of-127023.html
if-you-divide-7-131-by-5-which-remainder-do-you-get-83350.html
if-n-is-a-positive-integer-what-is-the-remainder-when-105067.html
what-is-the-unit-s-digit-of-101015.html
if-n-is-a-positive-integer-what-is-the-remainder-when-96262.html
what-is-the-remainder-when-3-243-is-divided-by-141050.html
find-the-ones-digit-of-141071.html
m12-72970.html

700+


when-51-25-is-divided-by-13-the-remainder-obtained-is-130220.html
m25-73474.html
which-of-the-following-numbers-is-prime-168179.html
what-is-the-remainder-when-43-86-is-divided-by-134778.html
when-51-25-is-divided-by-13-the-remainder-obtained-is-130220.html
what-is-the-remainder-when-43717-43628232-is-divided-by-154889.html
if-n-33-43-43-33-what-is-the-units-digit-of-n-140037.html

750+


what-is-the-units-digit-of-126681.html
what-is-the-remainder-of-126493.html
what-is-the-remainder-when-32-32-32-is-divided-by-100316.html
algebra-m26-145109.html
what-is-the-remainder-when-333-222-is-divided-by-156379.html
what-is-the-remainder-when-18-22-10-is-divided-by-99724.html


Data Sufficiency:


600+


if-k-is-a-positive-integer-what-is-the-reminder-when-2-k-126478.html

650+


if-x-is-a-positive-integer-is-the-remainder-0-when-3-x-109075.html
if-243-x-463-y-n-where-x-and-y-are-positive-integers-102054.html
if-r-s-and-t-are-all-positive-integers-what-is-the-136746.html

700+


tough-and-tricky-exponents-and-roots-questions-125967.html#p1029239
if-x-and-y-are-positive-integers-what-is-the-remainder-when-109636.html

Theory on Number Properties: math-number-theory-88376.html

All DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
All PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59

Theory on remainders problems: remainders-144665.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html


Hope this 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

Most Helpful Community Reply
Manager
Manager
avatar
Joined: 25 Jun 2012
Posts: 62
Location: India
WE: General Management (Energy and Utilities)
Re: If n = (33)^43  [#permalink]

Show Tags

New post 03 Oct 2012, 23:33
13
12
Archit143 wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?

0
2
4
6
8

pls suggest approach for such probs


In such type of exponents problem, first find the cyclicity of unit digit of 3. which is (3,9,7,1)

Now convert exponent value in to multiple of 4.
here 43=4(10)+3 so take 3^3 as unit value of first term which is 27 ===> unit digit is 7 (3,9,7,1)
Now 33=4(8)+1,so take 3^1 as unit value of second term which is 3^1====> unit digit is 3 (3,9,7,1)

Now unit digit of n = unit digit of first term + unit digit of 2nd term = 7+3=10 = unit digit is 0
General Discussion
Manager
Manager
User avatar
Joined: 08 Apr 2012
Posts: 118
Re: If n = (33)^43  [#permalink]

Show Tags

New post 03 Oct 2012, 23:41
9
Archit143 wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?

0
2
4
6
8

pls suggest approach for such probs


Hi Archit,

Just follow the unit digits:

n = (33)^43 + (43)^33

Here the last digits would determine the power.

So,

n = (33)^43 + (43)^33 ~ 3^43 + 3 ^33

Now last digits of 3^1, 2, 3, 4, 5 = 3, 9, 7, 1, 3. The cycle repeats after every 4 rounds.

So n = 3^43 + 3 ^33 = 3^(40+3) + 3^(32+1) = {(3^(4*10)}{3^3} + {(3^(4*8)}{3^1}

Now last digits of these terms would be {1}{7} + {1}{3} = 10

Hence the last digit is 0.

Hope this helps.

Regards,

Shouvik
_________________

Shouvik
http://www.Edvento.com
admin@edvento.com

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51218
Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

Show Tags

New post 04 Oct 2012, 03:27
4
1
Archit143 wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?

A. 0
B. 2
C. 4
D. 6
E. 8


Topic moved to PS forum.

Check Number Theory chapter of our Math Book to know how to deal with such problems: math-number-theory-88376.html (LAST DIGIT OF A POWER part)
_________________

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

SVP
SVP
User avatar
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1825
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

Show Tags

New post 26 Feb 2014, 19:16
2
Both terms has 3 at the units place, so the units place cycle around 3,9,7,1
33^43 will have 7 & 43^33 will have 3

7 + 3 = 10, 0 would be in the units place of the resultant

Answer = 0 = A
_________________

Kindly press "+1 Kudos" to appreciate :)

Intern
Intern
avatar
Joined: 25 Jul 2013
Posts: 1
GMAT ToolKit User
Re: If n = (33)^43  [#permalink]

Show Tags

New post 09 Mar 2014, 10:43
bhavinshah5685 wrote:
Archit143 wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?

0
2
4
6
8

pls suggest approach for such probs


In such type of exponents problem, first find the cyclicity of unit digit of 3. which is (3,9,7,1)

Now convert exponent value in to multiple of 4.
here 43=4(10)+3 so take 3^3 as unit value of first term which is 27 ===> unit digit is 7 (3,9,7,1)
Now 33=4(8)+1,so take 3^1 as unit value of second term which is 3^1====> unit digit is 3 (3,9,7,1)

Now unit digit of n = unit digit of first term + unit digit of 2nd term = 7+3=10 = unit digit is 0


I can't understand why it has to be a multiple of 4.
I mean, 43 can also be expressed as (2)(21)+1 but this would be wrong.
Generally in such exercises it has to be a multiple of 4, or I don't get something?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51218
Re: If n = (33)^43  [#permalink]

Show Tags

New post 09 Mar 2014, 11:56
7
8
Chrysopigi89 wrote:
bhavinshah5685 wrote:
Archit143 wrote:
If n = (33)^43 + (43)^33 what is the units digit of n?

0
2
4
6
8

pls suggest approach for such probs


In such type of exponents problem, first find the cyclicity of unit digit of 3. which is (3,9,7,1)

Now convert exponent value in to multiple of 4.
here 43=4(10)+3 so take 3^3 as unit value of first term which is 27 ===> unit digit is 7 (3,9,7,1)
Now 33=4(8)+1,so take 3^1 as unit value of second term which is 3^1====> unit digit is 3 (3,9,7,1)

Now unit digit of n = unit digit of first term + unit digit of 2nd term = 7+3=10 = unit digit is 0


I can't understand why it has to be a multiple of 4.
I mean, 43 can also be expressed as (2)(21)+1 but this would be wrong.
Generally in such exercises it has to be a multiple of 4, or I don't get something?


If n = (33)^43 + (43)^33 what is the units digit of n?

A. 0
B. 2
C. 4
D. 6
E. 8

First of all, the units digit of (33)^43 is the same as that of 3^43 and the units digit of (43)^33 is the same as that of 3^33. So, we need to find the units digit of 3^43 + 3^33.

Next, the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}:
3^1=3 (the units digit is 3)
3^2=9 (the units digit is 9)
3^3=27 (the units digit is 7)
3^4=81 (the units digit is 1)
3^5=243 (the units digit is 3 again!)
...

Thus:
The units digit of 3^43 is the same as the units digit of 3^3, so 7 (43 divided by the cyclicity of 4 gives the remainder of 3).
The units digit of 3^33 is the same as the units digit of 3^1, so 3 (33 divided by the cyclicity of 4 gives the remainder of 1).

Therefore the units digit of (33)^43 + (43)^33 is 7 + 3 = 0.

Answer: A.
_________________

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

Senior Manager
Senior Manager
User avatar
Status: Verbal Forum Moderator
Joined: 17 Apr 2013
Posts: 482
Location: India
GMAT 1: 710 Q50 V36
GMAT 2: 750 Q51 V41
GMAT 3: 790 Q51 V49
GPA: 3.3
Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

Show Tags

New post 09 Sep 2014, 01:46
Bunuel,

I found mistake in the solution given by you-

The Cycle here is

9, 7, 1, 3 Not 3, 9, 7 , 1 (as you stated)

Multiply 33 x 33 Once, what do we get 1089, Unit digit is 9. So the cycle should start at 9

By that logic we should add

1 + 9 = 10 Answer is still A.

But lets tweak it-

(33)^43 + (43)^34

Bunuel you will do this with your chosen cycle- 7+9 = 16, Your answer will be 6

With my chosen Cycle- 1+7 = 8.

That could be a trap if they play on the starting point of the cycle.
_________________

Like my post Send me a Kudos :) It is a Good manner.
My Debrief: http://gmatclub.com/forum/how-to-score-750-and-750-i-moved-from-710-to-189016.html

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51218
Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

Show Tags

New post 09 Sep 2014, 06:13
honchos wrote:
Bunuel,

I found mistake in the solution given by you-

The Cycle here is

9, 7, 1, 3
Not 3, 9, 7 , 1 (as you stated)

Multiply 33 x 33 Once, what do we get 1089, Unit digit is 9. So the cycle should start at 9

By that logic we should add

1 + 9 = 10 Answer is still A.

But lets tweak it-

(33)^43 + (43)^34

Bunuel you will do this with your chosen cycle- 7+9 = 16, Your answer will be 6

With my chosen Cycle- 1+7 = 8.

That could be a trap if they play on the starting point of the cycle.


Please re-read the solution and follow the links given above.

You should start with power of 1:
3^1=3 (the units digit is 3)
3^2=9 (the units digit is 9)
3^3=27 (the units digit is 7)
3^4=81 (the units digit is 1)
3^5=243 (the units digit is 3 again!)
...
_________________

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: 09 Feb 2014
Posts: 8
GMAT ToolKit User
If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

Show Tags

New post 05 Oct 2014, 04:40
2
2
Based on the Number Theory > Last digit of a power (in the GMATclub forum, I cannot link to the post)

Step 1: Find the last digit of 33^43
    find the cyclicity of 33
    33^1= _3
    33^2= _9
    33^3= _7
    33^4= _1
    33^5= _3
    -> 33 has the cyclicity of 4

      Divide the power by the cyclicity: 43/4 -> remainder is 3 (refer to the 3rd position in the cyclicity)
      -> the last digit of 33^43 is the same as that of 33^3 -> 7 is the last digit

      Step 2: Similarly, find the last digit of 43^33
      43 also has the cyclicity of 4 (ends with 3)
      Divide 33 by 4 -> remainder is 1 -> refer to the 1st position in the cyclicity -> last digit is 3

      Step 3: 7+3 = 10 -> the last digit is 0

      Answer A
      Manager
      Manager
      avatar
      Joined: 07 Apr 2015
      Posts: 165
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 10 May 2015, 00:50
      Bunuel wrote:
      Thus:
      The units digit of 3^43 is the same as the units digit of 3^3, so 7 (43 divided by the cyclicity of 4 gives the remainder of 3).
      The units digit of 3^33 is the same as the units digit of 3^1, so 3 (33 divided by the cyclicity of 4 gives the remainder of 1).

      Therefore the units digit of (33)^43 + (43)^33 is 7 + 3 = 0.

      Answer: A.


      I do not understand this part with remainder and cyclicity, why do i need to divide 43 by 4 to get a remainder of one which is then the units digit??
      Math Expert
      User avatar
      V
      Joined: 02 Sep 2009
      Posts: 51218
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 11 May 2015, 02:03
      1
      noTh1ng wrote:
      Bunuel wrote:
      Thus:
      The units digit of 3^43 is the same as the units digit of 3^3, so 7 (43 divided by the cyclicity of 4 gives the remainder of 3).
      The units digit of 3^33 is the same as the units digit of 3^1, so 3 (33 divided by the cyclicity of 4 gives the remainder of 1).

      Therefore the units digit of (33)^43 + (43)^33 is 7 + 3 = 0.

      Answer: A.


      I do not understand this part with remainder and cyclicity, why do i need to divide 43 by 4 to get a remainder of one which is then the units digit??


      You take the remainder obtained by dividing the power by cyclicity. For example, 3^43 --> 43 (power) divided by the cyclicity of 4 gives the remainder of 3 --> the units digit of 3^43 is the same as the units digit of 3^3.

      Check Units digits, exponents, remainders problems directory in our Special Questions Directory.
      _________________

      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
      User avatar
      Joined: 10 Jun 2015
      Posts: 118
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 14 Aug 2015, 21:38
      Archit143 wrote:
      If n = (33)^43 + (43)^33 what is the units digit of n?

      A. 0
      B. 2
      C. 4
      D. 6
      E. 8

      the unit digit of
      3^1=3
      3^5=3
      3^9=3
      3^33=3
      therefore, 3^33+3^33=6
      the correct answer is D
      Target Test Prep Representative
      User avatar
      P
      Status: Founder & CEO
      Affiliations: Target Test Prep
      Joined: 14 Oct 2015
      Posts: 4295
      Location: United States (CA)
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 23 Jun 2016, 08:33
      2
      1
      Archit143 wrote:
      If n = (33)^43 + (43)^33 what is the units digit of n?

      A. 0
      B. 2
      C. 4
      D. 6
      E. 8


      This is a units digit pattern question. The first thing to recognize is that in units digit pattern questions we only care about the units digit place value. Thus, we can rewrite the problem as:

      (3)^43 + (3)^33

      We now need to determine the units digit of (3)^43 + (3)^33. Let's determine the pattern of units digits that we get when a base of 3 is raised to consecutive exponents.

      3^1 = 3

      3^2 = 9

      3^3 = 27 (units digit of 7)

      3^4 = 81 (units digit of 1)

      3^5 = 243 (units digit of 3)

      Notice at 3^5, the pattern has started over:

      3^6 = units digit of 9

      3^7 = units digit of 7
      3^8 = units digit of 1

      So we can safely say that the base of 3 gives us a units digit pattern of 3, 9, 7, 1, 3, 9, 7, 1, …) that repeats every four exponents. Also notice that every time 3 is raised to an exponent that is a multiple of 4, we are left with a units digit of 1. This is very powerful information, which we can use to solve the problem. Let’s start with the units digit of (3)^43.

      An easy way to determine the units digit of (3)^43, is to find the closest multiple of 4 to 43, and that is 44. Thus we know:

      3^44 = units digit of 1

      So we can move back one exponent in our pattern and we get:

      3^43 = units digit of 7

      Let’s now determine the units digit of (3)^33.

      We already know that the pattern of units digits for powers of 3 will be 3, 9, 7, 1, 3, 9, 7, 1, … An easy way to determine the units digit of (3)^33 is to find the closest multiple of 4 to 33, and that is 32. Thus we know:

      3^32 = units digit of 1

      So we can move up one exponent in our pattern and we get:

      3^33 = units digit of 3

      The last step is to add the two units digits together so we have:

      7 + 3 = 10, which has a units digit of zero)

      Answer is A.
      _________________

      Scott Woodbury-Stewart
      Founder and CEO

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

      Director
      Director
      avatar
      G
      Joined: 02 Sep 2016
      Posts: 681
      Premium Member
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 01 Apr 2017, 05:07
      3
      Units digit of a product will ALWAYS depend on the units digits of the numbers being multiplied.

      Just focus on the units digit.
      Units digit of n=(3)^43+(3)^33
      The cyclicity of 3 is 4.
      3^1=3
      3^2=9
      3^3=27 (Units digit 7)
      3^4=81 (Units digit 1)

      After this the digits will start repeating (3,9,7,1,3,9,7,1, and so on)
      A the cyclicity is 4 (here), write the powers as multiple of 4 and look for the remainder (if any).

      43=4*10+3
      Therefore the units digit of (3)^43= 7

      33=4*8+1

      Therefore the units digit of (3)^33= 3

      The units digit of n= 7+3=10 (Just consider the last digit i.e. the unit's digit)
      The answer is 0.
      _________________

      Help me make my explanation better by providing a logical feedback.

      If you liked the post, HIT KUDOS !!

      Don't quit.............Do it.

      Intern
      Intern
      avatar
      B
      Joined: 11 Aug 2017
      Posts: 26
      Location: United States
      Schools: Duke '21 (I)
      GPA: 3.4
      Premium Member CAT Tests
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 29 Dec 2017, 14:02
      How is this only a 600 to 700 level question? Manhattan GMAT has it listed as "Devilish" under the difficulty.
      Math Expert
      User avatar
      V
      Joined: 02 Sep 2009
      Posts: 51218
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 29 Dec 2017, 23:03
      atr0038 wrote:
      How is this only a 600 to 700 level question? Manhattan GMAT has it listed as "Devilish" under the difficulty.


      The difficulty level of a question is calculated automatically based on the timer stats from the users which attempted the question.
      _________________

      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

      VP
      VP
      User avatar
      D
      Joined: 09 Mar 2016
      Posts: 1234
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 30 Dec 2017, 07:18
      bhavinshah5685 wrote:
      Archit143 wrote:
      If n = (33)^43 + (43)^33 what is the units digit of n?

      0
      2
      4
      6
      8

      pls suggest approach for such probs


      In such type of exponents problem, first find the cyclicity of unit digit of 3. which is (3,9,7,1)

      Now convert exponent value in to multiple of 4.
      here 43=4(10)+3 so take 3^3 as unit value of first term which is 27 ===> unit digit is 7 (3,9,7,1)
      Now 33=4(8)+1,so take 3^1 as unit value of second term which is 3^1====> unit digit is 3 (3,9,7,1)

      Now unit digit of n = unit digit of first term + unit digit of 2nd term = 7+3=10 = unit digit is 0


      Hello bhavinshah5685

      if exponent 33=4(8)+1 and unit digit is 3^1 than why did you highligh 1 in red instead of 3 ?

      thanks! :)
      e-GMAT Representative
      User avatar
      D
      Joined: 04 Jan 2015
      Posts: 2313
      Re: If n = (33)^43 + (43)^33 what is the units digit of n?  [#permalink]

      Show Tags

      New post 31 Dec 2017, 10:06
      dave13 wrote:
      bhavinshah5685 wrote:
      Archit143 wrote:
      If n = (33)^43 + (43)^33 what is the units digit of n?

      0
      2
      4
      6
      8

      pls suggest approach for such probs


      In such type of exponents problem, first find the cyclicity of unit digit of 3. which is (3,9,7,1)

      Now convert exponent value in to multiple of 4.
      here 43=4(10)+3 so take 3^3 as unit value of first term which is 27 ===> unit digit is 7 (3,9,7,1)
      Now 33=4(8)+1,so take 3^1 as unit value of second term which is 3^1====> unit digit is 3 (3,9,7,1)

      Now unit digit of n = unit digit of first term + unit digit of 2nd term = 7+3=10 = unit digit is 0


      Hello bhavinshah5685

      if exponent 33=4(8)+1 and unit digit is 3^1 than why did you highligh 1 in red instead of 3 ?

      thanks! :)



      He must have done it by mistake and highlighted 1 after getting the remainder as 1.

      By the way, his approach and answer are absolutely correct. :)


      Regards,
      Saquib
      e-GMAT
      Quant Expert
      _________________








      Register for free sessions
      Number Properties | Algebra |Quant Workshop

      Success Stories
      Guillermo's Success Story | Carrie's Success Story

      Ace GMAT quant
      Articles and Question to reach Q51 | Question of the week

      Must Read Articles
      Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
      Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
      Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
      Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
      Algebra- Wavy line | Inequalities

      Practice Questions
      Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets



      | '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

      GMAT Club Bot
      Re: If n = (33)^43 + (43)^33 what is the units digit of n? &nbs [#permalink] 31 Dec 2017, 10:06

      Go to page    1   2    Next  [ 21 posts ] 

      Display posts from previous: Sort by

      If n = (33)^43 + (43)^33 what is the units digit of n?

        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®.