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

It is currently 23 Jul 2018, 06:53

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

If a and b are positive integers, what is the remainder when 9^2a+b

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

Hide Tags

2 KUDOS received
Manager
Manager
User avatar
S
Joined: 17 Jan 2017
Posts: 61
Premium Member
If a and b are positive integers, what is the remainder when 9^2a+b  [#permalink]

Show Tags

New post 05 Apr 2018, 06:32
2
1
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

62% (00:45) correct 38% (01:21) wrong based on 58 sessions

HideShow timer Statistics

If a and b are positive integers, what is the remainder when \(9^{2a+b}\) is divided by 20?

(1) a = 3
(2) b = 5

Source: ExpertsGlobal
1 KUDOS received
PS Forum Moderator
avatar
D
Joined: 25 Feb 2013
Posts: 1185
Location: India
GPA: 3.82
GMAT ToolKit User Premium Member Reviews Badge
Re: If a and b are positive integers, what is the remainder when 9^2a+b  [#permalink]

Show Tags

New post 05 Apr 2018, 11:04
1
benejo wrote:
if a and b are positive integers, what is the remainder when \(9^{2a+b}\) is divided by 20?

(1) a = 3
(2) b = 5

Source: ExpertsGlobal


\(9^{2a+b}\) to know the remainder of this number when it is divided by 20 we need to know the last two digits of this number, i.e. we need to know the units and tens digit of this number.

The cyclicity of \(9\) is as follows -

\(9^1=9; 9^2=81; 9^3=729; 9^4=6561; 9^5=59049\)

So units digit of the number will be either \(1\) or \(9\) and tens digit will always be an even number. Hence the remainder of \(9^{2a+b}\) when divided by \(20\) will depend on the UNITS digit of this number.

Now \(2a\) is always EVEN irrespective of the value of \(a\). Hence \(9^{{Even}+b}\). So we only need to know the value of \(b\) to get the units digit.

Statement 1: Nothing mentioned about \(b\). Insufficient

Statement 2: \(b\) is odd so \(9^{2a+b}=9^{{Even}+Odd}=9^{Odd}\). Hence the remainder will be \(9\). Sufficient

Option B
Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47219
Re: If a and b are positive integers, what is the remainder when 9^2a+b  [#permalink]

Show Tags

New post 05 Apr 2018, 11:50
benejo wrote:
if a and b are positive integers, what is the remainder when \(9^{2a+b}\) is divided by 20?

(1) a = 3
(2) b = 5

Source: ExpertsGlobal


Similar questions to practice:
https://gmatclub.com/forum/if-a-and-b-a ... 28194.html
https://gmatclub.com/forum/if-a-and-b-a ... 36733.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: 07 Feb 2018
Posts: 3
Re: If a and b are positive integers, what is the remainder when 9^2a+b  [#permalink]

Show Tags

New post 12 Apr 2018, 22:13
Please explain to me why we can make sure that 9^odd (I know that the unit digit of which is always 9) when divided by 20 doesn't leave a remainder of 19 instead of 9? Thanks!
Expert Post
1 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47219
Re: If a and b are positive integers, what is the remainder when 9^2a+b  [#permalink]

Show Tags

New post 13 Apr 2018, 00:00
1
vap311096 wrote:
If a and b are positive integers, what is the remainder when \(9^{2a+b}\) is divided by 20?

(1) a = 3
(2) b = 5

Please explain to me why we can make sure that 9^odd (I know that the unit digit of which is always 9) when divided by 20 doesn't leave a remainder of 19 instead of 9? Thanks!


When x9 is divided by 20, the remainder could be either 9 or 19.

If x is odd, then x9 is 19, 39, 59, ..., all of which give the remainder of 19 upon division by 20.
If x is even, then x9 is 09, 29, 49, ..., all of which give the remainder of 9 upon division by 20.

Now, 9^odd, not only results in the units digit of 9 but also in the even tens digit: 9^1 = 09, 9^3 = 729, 9^5 = 59,049, ... Thus, 9^odd = ...(even)(9), which as we saw above will always give the remainder of 9 upon division by 20.

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?
Extra-hard Quant Tests with Brilliant Analytics

Re: If a and b are positive integers, what is the remainder when 9^2a+b &nbs [#permalink] 13 Apr 2018, 00:00
Display posts from previous: Sort by

If a and b are positive integers, what is the remainder when 9^2a+b

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

Events & Promotions

PREV
NEXT


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®.