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

It is currently 20 Aug 2018, 02:08

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 3^10–n is divisible by 4, which of the following could be the value

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

Hide Tags

Intern
Intern
avatar
B
Joined: 07 Jun 2018
Posts: 5
CAT Tests
If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post Updated on: 01 Aug 2018, 01:28
1
9
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

72% (01:11) correct 28% (01:12) wrong based on 182 sessions

HideShow timer Statistics

If \(3^{10}–n\) is divisible by 4, which of the following could be the value of an integer n?

I. 0
II. 1
III. 5

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

Originally posted by sharank on 10 Jul 2018, 15:28.
Last edited by Bunuel on 01 Aug 2018, 01:28, edited 3 times in total.
Added the topic name and formatted the question
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 6556
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 10 Jul 2018, 18:58
1
1
sharank wrote:
) If 3^10–n is divisible by 4, which of the following could be the value of an integer n?
I. 0 II. 1 III. 5

A. I only B. II only C. III only D. II and III only E. I, II and III

Posted from my mobile device


1) choices
The choices give you the answer..
3^10-n
I. If n is 0, we are left with 3^10, whi h is ODD, so not div by 4
Ii. Now 1 and 5 have a difference of 4 within themselves so either both are possible or none of the two

In choices D gives both and there is no choice giving None of the above
So D


2) cyclicity
3 gives a remainder 3
3^2=9 gives a remainder 1
3^3 =27 gives a remainder 3
3^4 =81 gives a remainder 1..
So even power leave a remainder 1
So 3^10 will leave a remainder 1 so n can be 1
Now 5 also leaves a remainder 1 so 5 can also be the answer..
D
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

RC Moderator
User avatar
P
Status: Perfecting myself for GMAT
Joined: 22 May 2017
Posts: 446
Concentration: Nonprofit
Schools: Haas '21
GPA: 4
WE: Engineering (Computer Software)
GMAT ToolKit User Premium Member CAT Tests
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 10 Jul 2018, 19:34
sharank wrote:
If 3^10–n is divisible by 4, which of the following could be the value of an integer n?



Is it \(3^{10-n}\) or \(3^{10}-n\)
_________________

If you like my post press kudos +1

New - RC Butler - 2 RC's everyday

Tag me in RC questions if you need help. Please provide your analysis of the question in the post along with the tag.

Manager
Manager
User avatar
B
Joined: 21 Jul 2017
Posts: 143
Location: India
Concentration: Social Entrepreneurship, Leadership
GMAT 1: 650 Q47 V33
GPA: 4
WE: Project Management (Education)
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 10 Jul 2018, 19:35
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

Posted from my mobile device
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 6556
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 10 Jul 2018, 19:48
rever08 wrote:
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

Posted from my mobile device


3*10 would end in 0
But 3^10=3*3*3*3*...10times
Cyclicity
3^1 ends in 3
3^2 =9 ends in 9
3^3 = 27 ends in 7
3^4=27*3 ends in 1
3^5=81*3 ends in 3 and the cyclicity happens after every 4th number
So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9..
So 3^anything positive will always be ODD
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Manager
Manager
User avatar
B
Joined: 21 Jul 2017
Posts: 143
Location: India
Concentration: Social Entrepreneurship, Leadership
GMAT 1: 650 Q47 V33
GPA: 4
WE: Project Management (Education)
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 10 Jul 2018, 19:51
chetan2u wrote:
rever08 wrote:
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

Posted from my mobile device


3*10 would end in 0
But 3^10=3*3*3*3*...10times
Cyclicity
3^1 ends in 3
3^2 =9 ends in 9
3^3 = 27 ends in 7
3^4=27*3 ends in 1
3^5=81*3 ends in 3 and the cyclicity happens after every 4th number
So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9..
So 3^anything positive will always be ODD


Argh..what was I thinking??
Thanks mate.
Senior Manager
Senior Manager
User avatar
G
Joined: 14 Dec 2017
Posts: 455
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 10 Jul 2018, 21:54
1
sharank wrote:
If \(3^{10}–n\) is divisible by 4, which of the following could be the value of an integer n?
I. 0
II. 1
III. 5

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

Posted from my mobile device



Given \(3^{10}–n\) = \(4k\)

I. \(n = 0\), we get \(3^{10}–n\) = \(3^{10}\) = \(3^{4}*3^{4}*3^{2}\)

When \(3^{4}*3^{4}*3^{2}\) is divided by \(4\), we get remainder as \(1\). Hence Not Divisible.

\(n = 0\) is not possible.


II. \(n = 1\), we get \(3^{10}–n\) = \(3^{10}–1\)

we know from above \(3^{10}\) divided by \(4\) leaves a remainder of \(1\)

& \(1\) divided by \(4\) will leave a remainder of \(1\).

Hence \(3^{10}–1\) divided by 4 will leave a remainder \((1 - 1) = 0\). Hence divisible.

\(n = 1\) is possible.


III. \(n = 5\), we get \(3^{10}–n\) = \(3^{10}–5\)

we know from above \(3^{10}\) divided by \(4\) leaves a remainder of \(1\)

& \(5\) divided by \(4\) will leave a remainder of \(1\).

Hence \(3^{10}–5\) divided by 4 will leave a remainder \((1 - 1) = 0\). Hence divisible.

\(n = 5\) is possible.


Answer D.


Thanks,
GyM
Senior Manager
Senior Manager
User avatar
P
Joined: 12 Feb 2015
Posts: 379
Premium Member CAT Tests
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 19 Jul 2018, 09:29
If you don't subtract anything than 3^10-0 will be odd hence option 1 and 5 can be eliminated immediately.

Now 3^10 is a big number, say x. If either 1 or 5 is subtracted from a big number the both the numbers will be divisible by 4 if either one of them is divisible by 4. (e.g. 9-1 = 8 & 9-5=4; BOTH 8 & 4 are divisible by 4)

Since there is no option which says none of these then we can safely select option D with doing any calculations!! Hope this out of the box thinking was helpful & time saving :-)
_________________

"Please hit :thumbup: +1 Kudos if you like this post" :student_man:

_________________
Manish :geek:

"Only I can change my life. No one can do it for me"

Senior Manager
Senior Manager
avatar
S
Joined: 15 Jan 2017
Posts: 367
CAT Tests
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 01 Aug 2018, 01:26
3^10 - n/ 4 = Q

So 3's cycle of power is 3,9,7,1...if we raise 3 to 10 times we will get a number that ends in XX9.
So 9 - 5 = 4 = divisible
9 - 8 = divisible.

So answer is II,III which is D
Senior Manager
Senior Manager
User avatar
G
Joined: 04 Sep 2015
Posts: 473
Location: India
WE: Information Technology (Computer Software)
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

New post 09 Aug 2018, 04:38
3=3
3*3 =9
3*3*3=27
3*3*3*3=81

then the last digit repeats itself.
so starting from 3 count 10 times we come to 9(last digit)

then we know that the last digit for 3^10 is 9.
and this -n is divisible by 4.that means n will be 1 or 5.
Re: If 3^10–n is divisible by 4, which of the following could be the value &nbs [#permalink] 09 Aug 2018, 04:38
Display posts from previous: Sort by

If 3^10–n is divisible by 4, which of the following could be the value

  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®.