Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 18 Jul 2019, 06:22

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

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

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

Author Message
TAGS:

Hide Tags

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

Show Tags

Updated on: 01 Aug 2018, 01:28
1
15
00:00

Difficulty:

35% (medium)

Question Stats:

70% (01:33) correct 30% (01:44) wrong based on 296 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
Joined: 02 Aug 2009
Posts: 7764
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

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
_________________
MBA Section Director
Affiliations: GMATClub
Joined: 22 May 2017
Posts: 2558
GPA: 4
WE: Engineering (Computer Software)
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

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 $$3^{10}-n$$
_________________
Manager
Joined: 21 Jul 2017
Posts: 190
Location: India
GMAT 1: 660 Q47 V34
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

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
Joined: 02 Aug 2009
Posts: 7764
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

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
_________________
Manager
Joined: 21 Jul 2017
Posts: 190
Location: India
GMAT 1: 660 Q47 V34
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

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.
Director
Joined: 14 Dec 2017
Posts: 517
Location: India
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

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.

Thanks,
GyM
_________________
Director
Joined: 12 Feb 2015
Posts: 875
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

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 +1 Kudos if you like this post"

_________________
Manish

"Only I can change my life. No one can do it for me"
Senior Manager
Joined: 15 Jan 2017
Posts: 348
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

01 Aug 2018, 01:26
1
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
Director
Joined: 04 Sep 2015
Posts: 627
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

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.
Manager
Joined: 29 May 2017
Posts: 127
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

10 Sep 2018, 00:50
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

can we reason as follows:

since 3^10 is odd, 3^10 -1 is even (odd-odd gives even). and since we get an even result, it could be divisible either by 4 or 5 ?

thanks
Manager
Joined: 20 Jul 2018
Posts: 75
WE: Corporate Finance (Investment Banking)
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

10 Sep 2018, 15:41
3^10=729^2= 29*29 = a number ending with 41. Therefore:
1) 41-0 = 41 not divisible by 4
2) 41-1 = 40 divisible by 4
3) 41-5 = 36 divisible by 4
CEO
Joined: 17 Nov 2007
Posts: 3372
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

Show Tags

10 Sep 2018, 16:30
3^10 = (4-1)^10 = 4k+1, where k is an integer.
So any n = 4m+1 will work, where m is an integer. Only 1 and 5 can be represented as 4m+1.

Posted from my mobile device
_________________
HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | Limited GMAT/GRE Math tutoring in Chicago
Re: If 3^10–n is divisible by 4, which of the following could be the value   [#permalink] 10 Sep 2018, 16:30
Display posts from previous: Sort by