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

 It is currently 11 Dec 2019, 11:46

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

What is the remainder when 2^86 is divided by 9?

Author Message
TAGS:

Hide Tags

Intern
Joined: 22 Feb 2014
Posts: 24
What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

Updated on: 16 Apr 2014, 01:58
2
41
00:00

Difficulty:

45% (medium)

Question Stats:

64% (01:31) correct 36% (01:54) wrong based on 533 sessions

HideShow timer Statistics

What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

Originally posted by GGMAT760 on 16 Apr 2014, 00:19.
Last edited by Bunuel on 16 Apr 2014, 01:58, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
MBA Section Director
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 7330
City: Pune
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

16 Apr 2014, 03:31
5
7
Rule :- Expression $$\frac{A*B*C*D*E}{K}$$ will give the same remainder as the expression $$\frac{Ar*Br*Cr*Dr*Er}{K}$$ where Ar, Br, Cr, Dr, Er are the remainders when divided by K individually.

2^86 / 9 can be simplified as $$\frac{2*2*2*2.......86 times}{9}$$

2*2*2 = 8. Remainder of 8/9 is 1. We can form 28 such groups of (2*2*2) and every group will give us the remainder as 1. After forming 28 groups we are left with 2*2 which when divided by 9 will give the remainder as 4

Now apply the rule cited above

Remainder of 2^86/9 is the same as remainder of $$\frac{1*1*1*......28 times * 4}{9}$$ ----------> $$\frac{4}{9}$$ ---------> 4

Let me know if anything still unclear
_________________
2020 MBA Applicants: Introduce Yourself Here!

MBA Video Series - Video answers to specific components and questions about MBA applications.

2020 MBA Deadlines, Essay Questions and Analysis of all top MBA programs
Intern
Joined: 06 Nov 2013
Posts: 3
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

14 Jun 2014, 18:46
8
3
I used the Binomial Theory (which I admit I'm a little shaky on...)

1. We need to find a relationship between our dividend (2^86) and divisor (9)

2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28

We can expand this into binomial form as:

4*(9-1)^28

Every term will have a factor of 9 in it EXCEPT the last term (-1)^28 which is just equal to 1. Don't forget to distribute the 4 out, and voila, we have our remainder of 4*(1) = 4 therefore answer D.
General Discussion
MBA Section Director
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 7330
City: Pune
Re: Can anyone help me to solve this strategically?  [#permalink]

Show Tags

16 Apr 2014, 00:36
drkomal2000 wrote:
Question 2: What is the remainder when 2^86 is divided by 9?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

I think it is D

2^86 / 9 --------> 28 times (2^3) * (2^2) / 9

Remainder when 2^3 divided by 9 is 1 and remainder when 2^2 divided by 9 is 4

So we have that 1*1*1.......28 times * 4 / 9 ------> 4/9 -------> Remainder = 4

_________________
2020 MBA Applicants: Introduce Yourself Here!

MBA Video Series - Video answers to specific components and questions about MBA applications.

2020 MBA Deadlines, Essay Questions and Analysis of all top MBA programs
Manager
Joined: 04 Jan 2014
Posts: 94
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

16 Apr 2014, 03:11
Sorry I don't quite get it. Are we trying to factorize by breaking down into factors with same power (in this case 2^2 / 3^2)?
Intern
Joined: 06 Dec 2013
Posts: 5
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

06 Jun 2014, 02:30
1
I tackled this problem as the following:
First, notice that you are asked to deal with one digit number: 9; therefore the remainder is one digit number too(take a look at the answer choices).
From this it follows that this question is about units digit of 2^86.
Senior Manager
Joined: 08 Apr 2012
Posts: 323
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

08 Jun 2014, 02:56
3
1
Narenn wrote:
Rule :- Expression $$\frac{A*B*C*D*E}{K}$$ will give the same remainder as the expression $$\frac{Ar*Br*Cr*Dr*Er}{K}$$ where Ar, Br, Cr, Dr, Er are the remainders when divided by K individually.

2^86 / 9 can be simplified as $$\frac{2*2*2*2.......86 times}{9}$$

2*2*2 = 8. Remainder of 8/9 is 1. We can form 28 such groups of (2*2*2) and every group will give us the remainder as 1. After forming 28 groups we are left with 2*2 which when divided by 9 will give the remainder as 4

Now apply the rule cited above

Remainder of 2^86/9 is the same as remainder of $$\frac{1*1*1*......28 times * 4}{9}$$ ----------> $$\frac{4}{9}$$ ---------> 4

Let me know if anything still unclear

I'm sorry, but isn't 8/9 yield a remainder of 8?
Intern
Joined: 20 Nov 2013
Posts: 25
Schools: LBS '17
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

10 Jun 2014, 12:16
1
I am not sure if the way i approached it is correct.
The cyclicity of 2 is 4. So 86/4 gives us a remainder of 2 which means the units digit of 2^86 will have a units digit of 4.
Senior Manager
Joined: 13 Jun 2013
Posts: 255
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

10 Jun 2014, 12:43
1
amz14 wrote:
I am not sure if the way i approached it is correct.
The cyclicity of 2 is 4. So 86/4 gives us a remainder of 2 which means the units digit of 2^86 will have a units digit of 4.

This approach is not correct, because cyclicity will only tell you about the last digit and not about the reminder. consider this 2^10 also ends with 4 i.e. 1024, but when it is divided by 9 the remainder is not 4 but 7.
Intern
Joined: 21 Dec 2011
Posts: 14
Schools: Tepper '17 (S)
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

10 Jun 2014, 13:10
1
1
I did it the following way:
as 9*7 = 63, 64/63 will always give a remainder of 1. 64 is 2^6.
So the new number becomes,
[{2^(6*14)}*2^2]/9

=> {2^(6*14)}/9 will always give remainder 1.
2^2/9 will give e remainder of 4.
So my answer will be D(4)
Manager
Joined: 28 Dec 2013
Posts: 65
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

14 Jun 2014, 19:32
1
2
dcbark01 wrote:
I used the Binomial Theory (which I admit I'm a little shaky on...)

1. We need to find a relationship between our dividend (2^86) and divisor (9)

2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28

We can expand this into binomial form as:

4*(9-1)^28

Every term will have a factor of 9 in it EXCEPT the last term (-1)^28 which is just equal to 1. Don't forget to distribute the 4 out, and voila, we have our remainder of 4*(1) = 4 therefore answer D.

I really like this method, I have a question though, and my sincere apologies if its stupid. When you do this step :2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28, I see where the (2^2) * (2^84) is coming from but I get confused where 4*(2^3)^28 = 4*(8)^28 is coming from. Step by step where is 4*(2^3)^28 coming from? Is there distribution happening?
Intern
Joined: 28 Mar 2014
Posts: 19
Location: India
GPA: 3
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

14 Jun 2014, 19:37
5
When a power of 2 is divided by 9, the remainder start repeating itsel after 2^6 means from 2^1 to 2^6 the remainder goes like, 2,4,8,7,5,1 and then from 2^7 it start repeating itself. so if we divide 86 by 6 it gives 2 as remainder that means only 2*2 left which give 4 as remainder for the question.
Intern
Joined: 28 Mar 2014
Posts: 19
Location: India
GPA: 3
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

14 Jun 2014, 19:38
sagnik242 wrote:
dcbark01 wrote:
I used the Binomial Theory (which I admit I'm a little shaky on...)

1. We need to find a relationship between our dividend (2^86) and divisor (9)

2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28

We can expand this into binomial form as:

4*(9-1)^28

Every term will have a factor of 9 in it EXCEPT the last term (-1)^28 which is just equal to 1. Don't forget to distribute the 4 out, and voila, we have our remainder of 4*(1) = 4 therefore answer D.

I really like this method, I have a question though, and my sincere apologies if its stupid. When you do this step :2^86 = (2^2)*(2^84) = 4*(2^3)^28 = 4*(8)^28, I see where the (2^2) * (2^84) is coming from but I get confused where 4*(2^3)^28 = 4*(8)^28 is coming from. Step by step where is 4*(2^3)^28 coming from? Is there distribution happening?

Yes its distribution of powers. 3*28=84
Manager
Joined: 02 Sep 2014
Posts: 68
Location: United States
GMAT 1: 700 Q49 V37
GMAT 2: 700 Q47 V40
GMAT 3: 720 Q48 V41
GPA: 3.26
WE: Consulting (Consulting)
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

01 Nov 2014, 13:49
Can someone help me figure out the next step with the method I tried to use? I'm having a hard time understanding the other methods used in this thread.

I found that 2^86 would end with a 6, but I'm unsure what to do after that. If the question asked what the remainder would be if divided by 10, this would be an easier question.
Senior Manager
Joined: 10 Mar 2013
Posts: 461
Location: Germany
Concentration: Finance, Entrepreneurship
Schools: WHU MBA"20 (A\$)
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

24 Oct 2015, 11:48
1
GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

$$2^86$$ has a unit digit 4, 4/86 has a remainder 4 Answer (D)
CEO
Joined: 20 Mar 2014
Posts: 2560
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

24 Oct 2015, 12:09
2
2
BrainLab wrote:
GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

$$2^86$$ has a unit digit 4, 4/86 has a remainder 4 Answer (D)

Again, you are making the mistake of going for digits to come up with the answer.

Best way will be 2^84 = 2^2*(2^3)^28 = 4*(8)^28 = 4*(9-1)^28

All the terms in (9-1)^28 will be multiples of 9 except the last one (-1)^28 = 28.

Thus, you need to find the remainder when 4*(9-1)^28 is divided by 9 or 4*(-1)^28 is divided by 9 or 4 is divided by 9, giving you a remainder of 4.

Hope this helps.
Senior Manager
Joined: 07 Aug 2011
Posts: 499
GMAT 1: 630 Q49 V27
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

24 Oct 2015, 21:05
1
1
GGMAT760 wrote:
What is the remainder when 2^86 is divided by 9?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

we know that when 64 is divided by 9 , remainder is 1 .

2^86 = 2^2 * 2^84 = 4 * (2^6)^14 = 4 * 64^14

Now , remember that if X= A*B then R(X/K) = ( R(A/K) *R(B/K) ) /K for eg:
65/9 remainder is 2
65= 13*5
R(13/9) * R(5/9) = 4*5 = 20
and R(20/9) = 2

Back to original question now -
2^86 = 2^2 * 2^84 = 4 * (2^6)^14 = 4 * ( 64^14 )

4/9 = 4 remainder
64/9 =1 remainder

so 4 * ( 64^14 ) / 9 remainder = 4

Manager
Joined: 18 Jun 2017
Posts: 58
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

19 Jul 2017, 10:52
1
we know cyclicity of 2 is 4. Now on dividing 86/4 we get 2 remainder for which unit's place of 2^86 should be 2^2=4. Hence dividing unit's place digit by 9 leaves remainder as 4. Option D
Manager
Joined: 04 May 2014
Posts: 150
Location: India
WE: Sales (Mutual Funds and Brokerage)
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

20 Aug 2017, 23:23
8/9 remainder is 1? should it not be 8?

Narenn wrote:
Rule :- Expression $$\frac{A*B*C*D*E}{K}$$ will give the same remainder as the expression $$\frac{Ar*Br*Cr*Dr*Er}{K}$$ where Ar, Br, Cr, Dr, Er are the remainders when divided by K individually.

2^86 / 9 can be simplified as $$\frac{2*2*2*2.......86 times}{9}$$

2*2*2 = 8. Remainder of 8/9 is 1. We can form 28 such groups of (2*2*2) and every group will give us the remainder as 1. After forming 28 groups we are left with 2*2 which when divided by 9 will give the remainder as 4

Now apply the rule cited above

Remainder of 2^86/9 is the same as remainder of $$\frac{1*1*1*......28 times * 4}{9}$$ ----------> $$\frac{4}{9}$$ ---------> 4

Let me know if anything still unclear
Math Expert
Joined: 02 Sep 2009
Posts: 59674
Re: What is the remainder when 2^86 is divided by 9?  [#permalink]

Show Tags

21 Aug 2017, 01:59
gps5441 wrote:
8/9 remainder is 1? should it not be 8?

Yes, 8 divided by 9 yields the remainder of 8.
_________________
Re: What is the remainder when 2^86 is divided by 9?   [#permalink] 21 Aug 2017, 01:59

Go to page    1   2    Next  [ 37 posts ]

Display posts from previous: Sort by