November 17, 2018 November 17, 2018 07:00 AM PST 09:00 AM PST Nov. 17, 7 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. November 17, 2018 November 17, 2018 09:00 AM PST 11:00 AM PST Join the Quiz Saturday November 17th, 9 AM PST. The Quiz will last approximately 2 hours. Make sure you are on time or you will be at a disadvantage.
Author 
Message 
TAGS:

Hide Tags

Director
Status: No dream is too large, no dreamer is too small
Joined: 14 Jul 2010
Posts: 530

If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
Updated on: 20 Oct 2014, 07:35
Question Stats:
63% (01:38) correct 37% (01:50) wrong based on 870 sessions
HideShow timer Statistics
If k is a positive integer, What is the remainder when 2^k is divided by 10? (1) k is divisible by 10 (2) k is divisible by 4 My approach is as follows: (1) k could be 10, 20, 30... case i. if k = 10, 2^10, the cyclicity of 2 is 4 (10/4 = reminder 2) so 2^2 is divided by 10 and reminder is 4 case ii. if k = 20, 2^20, the cyclicity of 2 is 4 (20/4 = 5, 5/4 = reminder 1) so 2^1 is divided by 10 and reminder is 2 Insufficient.
(2) k = 4, 8, 12 2^4, the cyclicity of 2 is 4 (4/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 2^8, the cyclicity of 2 is 4 (8/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 Sufficient.
Ans. B
Please help whether the above approach can be applied in the problem?
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Collections: PSof OG solved by GC members: http://gmatclub.com/forum/collectionpswithsolutionfromgmatclub110005.html DS of OG solved by GC members: http://gmatclub.com/forum/collectiondswithsolutionfromgmatclub110004.html 100 GMAT PREP Quantitative collection http://gmatclub.com/forum/gmatprepproblemcollections114358.html Collections of work/rate problems with solutions http://gmatclub.com/forum/collectionsofworkrateproblemwithsolutions118919.html Mixture problems in a file with best solutions: http://gmatclub.com/forum/mixtureproblemswithbestandeasysolutionsalltogether124644.html
Originally posted by Baten80 on 23 Jan 2012, 20:55.
Last edited by Bunuel on 20 Oct 2014, 07:35, edited 2 times in total.
Edited the question.




Math Expert
Joined: 02 Sep 2009
Posts: 50619

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
24 Jan 2012, 01:05
Baten80 wrote: If k is a positive integer, What is the remainder when 2^k is divided by 10? 1) k is dividable by 10 2) k is dividable by 4
My approach is as follows: (1) k could be 10, 20, 30... case i. if k = 10, 2^10, the cyclicity of 2 is 4 (10/4 = reminder 2) so 2^2 is divided by 10 and reminder is 4 case ii. if k = 20, 2^20, the cyclicity of 2 is 4 (20/4 = 5, 5/4 = reminder 1) so 2^1 is divided by 10 and reminder is 2 Insufficient.
(2) k = 4, 8, 12 2^4, the cyclicity of 2 is 4 (4/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 2^8, the cyclicity of 2 is 4 (8/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 Sufficient.
Ans. B
Please help whether the above approach can be applied in the problem? General approach is correct, though the red parts are not. The last digit of 2^k repeats in pattern of 4 (cyclicity is 4): 2^1=2 > last digit is 2; 2^2=4 > last digit is 4; 2^3=8 > last digit is 8; 2^4=16 > last digit is 6;2^5=32 > last digit is 2 again; Now, when k itself is a multiple of 4 (when there is no remainder upon division k by cyclicity number), then the last digit will be the last digit of 2^4 (4th in pattern), so 6 not 1 (taking 2^0) as you've written. If k is a positive integer, what is the remainder when 2^k is divided by 10?Notice that all we need to know to answer the question is the last digit of 2^k. (1) k is divisible by 10 > different multiples of 10 yield different remainders upon division by 4 (for example 10/4 yields 2 and 20/4 yields 0), thus we can not get the single numerical value of the last digit of 2^k. Not sufficient. (2) k is divisible by 4 > as discussed, when k is a multiple of 4, the last digit of 2^k equals to the last digit of 2^4, which is 6. Integer ending with 6 yields remainder of 6 upon division by 10. Sufficient. Answer: B. Hope it's clear.
_________________
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? Extrahard Quant Tests with Brilliant Analytics




Director
Status: No dream is too large, no dreamer is too small
Joined: 14 Jul 2010
Posts: 530

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
25 Jan 2012, 08:57



Math Expert
Joined: 02 Sep 2009
Posts: 50619

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
06 Jun 2013, 05:10



Manager
Joined: 02 Jul 2012
Posts: 191
Location: India
GPA: 2.6
WE: Information Technology (Consulting)

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
20 Oct 2014, 07:27
Baten80 wrote: If k is a positive integer, What is the remainder when 2^k is divided by 10?
(1) k is divisible by 10 (2) k is divisible by 4
2^k divided by 10. The cycliicity of 2 when divided by 10 is 4. 1  k is divisible by 10  the number can be 10 (2) or 20(0)  Not Sufficient 2  k is divisible by 4  Sufficient. Ans. B
_________________
Give KUDOS if the post helps you...



Manager
Joined: 26 May 2013
Posts: 95

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
20 Oct 2014, 08:31
Bunuel wrote: Baten80 wrote: If k is a positive integer, What is the remainder when 2^k is divided by 10? 1) k is dividable by 10 2) k is dividable by 4
My approach is as follows: (1) k could be 10, 20, 30... case i. if k = 10, 2^10, the cyclicity of 2 is 4 (10/4 = reminder 2) so 2^2 is divided by 10 and reminder is 4 case ii. if k = 20, 2^20, the cyclicity of 2 is 4 (20/4 = 5, 5/4 = reminder 1) so 2^1 is divided by 10 and reminder is 2 Insufficient.
(2) k = 4, 8, 12 2^4, the cyclicity of 2 is 4 (4/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 2^8, the cyclicity of 2 is 4 (8/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 Sufficient.
Ans. B
Please help whether the above approach can be applied in the problem? General approach is correct, though the red parts are not. The last digit of 2^k repeats in pattern of 4 (cyclicity is 4): 2^1=2 > last digit is 2; 2^2=4 > last digit is 4; 2^3=8 > last digit is 8; 2^4=16 > last digit is 6;2^5=32 > last digit is 2 again; Now, when k itself is a multiple of 4 (when there is no remainder upon division k by cyclicity number), then the last digit will be the last digit of 2^4 (4th in pattern), so 6 not 1 (taking 2^0) as you've written. If k is a positive integer, what is the remainder when 2^k is divided by 10?Notice that all we need to know to answer the question is the last digit of 2^k. (1) k is divisible by 10 > different multiples of 10 yield different remainders upon division by 4 (for example 10/4 yields 2 and 20/4 yields 0), thus we can not get the single numerical value of the last digit of 2^k. Not sufficient. (2) k is divisible by 4 > as discussed, when k is a multiple of 4, the last digit of 2^k equals to the last digit of 2^4, which is 6. Integer ending with 6 yields remainder of 6 upon division by 10. Sufficient. Answer: B. Hope it's clear. To add some clarity for myself and viewers: Since the last digit in 2^k repeats in cycles of 4, you will ALWAYS know the last digit (and remainder) if k is a multiple of 4. Therefore 2^4, 2^8,2^12. 2_16, etc.... will always have a last digit of 6. If k is a multiple of 10, you know if k = 10, the last digit will be 4, and if k=20 the last digit will be 6, k=30 the last digit will be 4, etc... in repeating pattern. However without knowing the exact value of k you won't know the remainder.



Manager
Joined: 22 Jan 2014
Posts: 176
WE: Project Management (Computer Hardware)

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
02 Nov 2014, 01:22
Baten80 wrote: If k is a positive integer, What is the remainder when 2^k is divided by 10? (1) k is divisible by 10 (2) k is divisible by 4 My approach is as follows: (1) k could be 10, 20, 30... case i. if k = 10, 2^10, the cyclicity of 2 is 4 (10/4 = reminder 2) so 2^2 is divided by 10 and reminder is 4 case ii. if k = 20, 2^20, the cyclicity of 2 is 4 (20/4 = 5, 5/4 = reminder 1) so 2^1 is divided by 10 and reminder is 2 Insufficient.
(2) k = 4, 8, 12 2^4, the cyclicity of 2 is 4 (4/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 2^8, the cyclicity of 2 is 4 (8/4 = reminder 0) so 2^0 is divided by 10 and reminder is 1 Sufficient.
Ans. B
Please help whether the above approach can be applied in the problem? remainder by 10 means units digit. 1) k is div by 10 k = 10 ; 2^10 ends in 4 k = 20 ; 2^20 ends in 6 insufficient. 2) k is div by 4 2^(4k) always ends in 6 sufficient. B.
_________________
Illegitimi non carborundum.



Current Student
Joined: 12 Aug 2015
Posts: 2633

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
22 Mar 2016, 01:47



NonHuman User
Joined: 09 Sep 2013
Posts: 8791

Re: If k is a positive integer, What is the remainder when 2^k is divided
[#permalink]
Show Tags
10 Jul 2018, 08:15
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




Re: If k is a positive integer, What is the remainder when 2^k is divided &nbs
[#permalink]
10 Jul 2018, 08:15






