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

It is currently 19 Feb 2019, 23:48

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 February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
  • Free GMAT Prep Hour

     February 20, 2019

     February 20, 2019

     08:00 PM EST

     09:00 PM EST

    Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST
  • Online GMAT boot camp for FREE

     February 21, 2019

     February 21, 2019

     10:00 PM PST

     11:00 PM PST

    Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

The Units Digits of Big Powers

  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: 53004
The Units Digits of Big Powers  [#permalink]

Show Tags

New post 04 Feb 2019, 01:58
3
1
THE UNITS DIGIT OF A POWER

How to Determine the Units (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 \(3\).

Find below links to units digits, exponents, remainders, cyclicity problems.

Problem Solving:


650+


http://gmatclub.com/forum/what-is-the-t ... 27023.html
http://gmatclub.com/forum/if-you-divide ... 83350.html
http://gmatclub.com/forum/if-n-is-a-pos ... 05067.html
http://gmatclub.com/forum/what-is-the-u ... 01015.html
http://gmatclub.com/forum/if-n-is-a-pos ... 96262.html
http://gmatclub.com/forum/what-is-the-r ... 41050.html
http://gmatclub.com/forum/find-the-ones ... 41071.html
http://gmatclub.com/forum/m12-72970.html
http://gmatclub.com/forum/if-a-and-b-ar ... 87301.html

700+


http://gmatclub.com/forum/when-51-25-is ... 30220.html
http://gmatclub.com/forum/m25-73474.html
http://gmatclub.com/forum/which-of-the- ... 68179.html
http://gmatclub.com/forum/what-is-the-r ... 34778.html
http://gmatclub.com/forum/what-is-the-r ... 54889.html
http://gmatclub.com/forum/if-n-33-43-43 ... 40037.html

750+


http://gmatclub.com/forum/what-is-the-u ... 26681.html
http://gmatclub.com/forum/what-is-the-r ... 26493.html
http://gmatclub.com/forum/what-is-the-r ... 00316.html
http://gmatclub.com/forum/algebra-m26-145109.html
http://gmatclub.com/forum/what-is-the-r ... 56379.html
http://gmatclub.com/forum/what-is-the-r ... 99724.html


Data Sufficiency:


600+


http://gmatclub.com/forum/if-k-is-a-pos ... 26478.html
http://gmatclub.com/forum/if-r-s-and-t- ... 87298.html

650+


http://gmatclub.com/forum/if-x-is-a-pos ... 09075.html
http://gmatclub.com/forum/if-243-x-463- ... 02054.html
http://gmatclub.com/forum/if-r-s-and-t- ... 36746.html
http://gmatclub.com/forum/if-x-and-y-ar ... 87302.html

700+


http://gmatclub.com/forum/tough-and-tri ... l#p1029239
http://gmatclub.com/forum/if-x-and-y-ar ... 09636.html
_________________

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
B
Joined: 04 Mar 2017
Posts: 7
Reviews Badge
Re: The Units Digits of Big Powers  [#permalink]

Show Tags

New post 04 Feb 2019, 17:41
I have learnt all except 7, seems useful for GMAT
GMAT Club Bot
Re: The Units Digits of Big Powers   [#permalink] 04 Feb 2019, 17:41
Display posts from previous: Sort by

The Units Digits of Big Powers

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


cron
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®.