NEW HERE? LEARN MORE
closeclose
Last visit was: 25 Apr 2024, 14:47 It is currently 25 Apr 2024, 14:47
Decision Tracker
My Rewards
New posts
New comers' posts

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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

What is the remainder when 5^91 is divided by 91?

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Chethan92
Director
Director
Joined: 18 Jul 2018
Posts: 926
Own Kudos [?]: 1288 [60]
Given Kudos: 95
Location: India
Concentration: Operations, General Management
Schools: IIMB EPGP'22 (M)
GMAT 1: 590 Q46 V25
GMAT 2: 690 Q49 V34
WE:Engineering (Energy and Utilities)
Send PM
Reviews Badge
What is the remainder when 5^91 is divided by 91? [#permalink] New post   06 Aug 2018, 09:53
1
Kudos
59
Bookmarks
00:00
A
B
C
D
E

Difficulty:

95% (hard)

Question Stats:

16% (02:10) correct 84% (02:14) wrong based on 304 sessions
Hide Show timer Statistics
What is the remainder when \(5^{91}\) is divided by 91?

A. 5
B. 8
C. 13
D. 34
E. 47
ShowHide Answer
Official Answer
E
Most Helpful Reply
User avatar
L
chetan2u
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11178
Own Kudos [?]: 31932 [10]
Given Kudos: 290
Send PM
Reviews Badge
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   06 Aug 2018, 19:32
5
Kudos
5
Bookmarks
Expert Reply
Afc0892 wrote:
What is the remainder when \(5^{91}\) is divided by 91?

1) 5
2) 8
3) 13
4) 34
5) 47



I doubt such a question will come in actuals..

Otherwise..
1) as per remainder theorem
\(91=7*13\) so LCM of (7-1,13-1) or (6,12) is 12
So \(5^{91}\) will give the same remainder as \(5^7\), since 91/12 gives 7 as remainder.
\(5^7=(5^3)^2*5=(125)^2*5\)
125 will leave 125-91=34 as remainder
So \(34*34*5=34*170\)
170 will leave a remainder of -12
\(34*(-12)= 68*(-6)=-23*-6=138\)
So remainder=138-91=47

2) if you do not know the theorem you will have to get the terms closer to multiple of 91
So \(5^{91}=(5^4)^{22}*5^3=625^{22}*5^3\)
91*7=637, so remainder is \(625-637=(-12)\)
So new term is \((-12)^{22}*5^3\)
As even power, it will be same as \(12^{22}*5^3=4^{22}*3^{22}*5^3\)

Also \(4^3=64\) and remainder is 64-91=-27
So \((4^3)^7*4*3^{22}*5^3=(-27)^7*3^{22}*5^3=(-3^3)^7*3^{22}*5^3=-(3^{43}*5^3*4)\)
Now \(3^6=729\) and 91*8=728 so remainder is 1
\(-(3^6)^7*3*5^3*4=-(1^7*3*5^3*4)=-1*3*5*100=-1*3*5*9=-135\)
So POSITIVE remainder will be \(91*2-135=182-135=47\)

E

you could have used this too to simplify
Now \(2^7*5=128*5=640\) remainder will be 640-637=3
avatar
B
rahulkashyap
Manager
Manager
Joined: 09 Oct 2015
Posts: 174
Own Kudos [?]: 61 [7]
Given Kudos: 28
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   06 Aug 2018, 22:03
3
Kudos
4
Bookmarks
91 = 7*13

Let us find out Rem[5^91/7] and Rem[5^91/13] - using binomial expansion
We will combine them later.

Rem [5^91/7]

= 5(5)^90 / 7
= 5( 125) ^ 30 / 7
=5 (126-1)^30 / 7

remainder = 5. (-1)^30 = 5 / 7
rem = 5



Rem [5^91/13]
= Rem 5. (5)^90 / 13
= 5. (25)^45 / 13
= 5. (26-1)^45)/ 13
remainder = 5. (-1)^45 / 13
= -5/13 = r= 8



So, our answer is a number which leaves a remainder of 5 when divided by 7 and it should leave a remainder of 8 when divided by 13.

simplifying this, we get N= 91K + 47
Hence 47 is the reaminder
General Discussion
User avatar
L
globaldesi
VP
VP
Joined: 28 Jul 2016
Posts: 1212
Own Kudos [?]: 1728 [2]
Given Kudos: 67
Location: India
Concentration: Finance, Human Resources
Schools: ISB '18 (D)
GPA: 3.97
WE:Project Management (Investment Banking)
Send PM
Reviews Badge
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   21 Feb 2020, 21:38
1
Kudos
1
Bookmarks
chetan2u wrote:
Afc0892 wrote:
What is the remainder when \(5^{91}\) is divided by 91?

1) 5
2) 8
3) 13
4) 34
5) 47



I doubt such a question will come in actuals..

Otherwise..
1) as per remainder theorem
\(91=7*13\) so LCM of (7-1,13-1) or (6,12) is 12
So \(5^{91}\) will give the same remainder as \(5^7\), since 91/12 gives 7 as remainder.
\(5^7=(5^3)^2*5=(125)^2*5\)
125 will leave 125-91=34 as remainder
So \(34*34*5=34*170\)
170 will leave a remainder of -12
\(34*(-12)= 68*(-6)=-23*-6=138\)
So remainder=138-91=47

2) if you do not know the theorem you will have to get the terms closer to multiple of 91
So \(5^{91}=(5^4)^{22}*5^3=625^{22}*5^3\)
91*7=637, so remainder is \(625-637=(-12)\)
So new term is \((-12)^{22}*5^3\)
As even power, it will be same as \(12^{22}*5^3=4^{22}*3^{22}*5^3\)

Also \(4^3=64\) and remainder is 64-91=-27
So \((4^3)^7*4*3^{22}*5^3=(-27)^7*3^{22}*5^3=(-3^3)^7*3^{22}*5^3=-(3^{43}*5^3*4)\)
Now \(3^6=729\) and 91*8=728 so remainder is 1
\(-(3^6)^7*3*5^3*4=-(1^7*3*5^3*4)=-1*3*5*100=-1*3*5*9=-135\)
So POSITIVE remainder will be \(91*2-135=182-135=47\)

E

you could have used this too to simplify
Now \(2^7*5=128*5=640\) remainder will be 640-637=3

chetan2u can you share some link for remainder theorem you used in 1)
User avatar
B
ShailendraSYadav
Intern
Intern
Joined: 15 Apr 2020
Posts: 19
Own Kudos [?]: 6 [2]
Given Kudos: 319
Location: India
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   22 Jul 2020, 04:23
1
Bookmarks
5^91/91...Remainder?

Solution:
=5^91/91
=5^91/7*13

=Points to observe
5 (num) and 91(deno) are co-primes
Need to factor the denominator,

=We need to develop 2 equations (as 2 factors are there in denom) based on divisibility of individual denominators..

Part:1
=5^91/7
=Based on Euler's theorem, Euler number for prime number at denom. will be 6. (7-1=6)
=dividing 91/6= 6K+1
=so 5^1/7= 5
=5 remainder.
=Equation1 N1= 7A+5

Part:2
=5^91/13
=Based on Euler's theorem, Euler number for prime number at denom. will be 12. (13-1=12)
=dividing 91/13= 12K+7
=so 5^7/13
=25*25*25*5/13
=8 Remainder
= Equation2 N2= 13B+8

Fill the values in Equation N1 (N1= 7A+5), Values are= 5,12,19,26,33,40,47
Fill the values in Equation N2 (N2= 13B+8), Values are= 8,21,34,47

First value satisfying both the equations will be remainder, i.e. 47.

Thanks!
User avatar
B
romasharma1998
Intern
Intern
Joined: 13 Jan 2022
Posts: 12
Own Kudos [?]: 1 [1]
Given Kudos: 192
Location: India
Schools: Booth '25 CEIBS Darden '25 IESE '25 ISB '24 Kellogg '25 LBS '25 NUS '25 SDA Bocconi "24 Yale '25
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   05 Jul 2022, 02:24
1
Kudos
Bunuel

can you help with this ? can such qs be asked in the gmat
User avatar
B
azooktutor
Intern
Intern
Joined: 28 Apr 2018
Posts: 15
Own Kudos [?]: 14 [0]
Given Kudos: 8
GMAT 1: 740 Q48 V42
GMAT 2: 750 Q49 V44
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   06 Aug 2018, 17:37
This one is really tricky! I don't have an answer yet, but I'm wondering if the fact that \(5^7\) does have a remainder of 47, and that \(5^{91} = (5^7)^{13}\), is significant.

I'm also wondering if 91's proximity to 100 could be useful. If you were to divide any power of 5 (over \(5^1\)) by 100, you would always get a remainder of 25. So perhaps the answer has something to do with using 100 as an approximation and then adding to the remainder.

Very interested to hear others' thoughts!

Edit: formatting
avatar
B
mimoh
Intern
Intern
Joined: 06 Jul 2021
Posts: 2
Own Kudos [?]: 0 [0]
Given Kudos: 16
Location: India
GMAT Focus 1:
645 Q86 V82 DI77
GMAT 1: 650 Q49 V30
GMAT 2: 650 Q49 V28
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   10 Jul 2022, 10:57
chetan2u wrote:
Afc0892 wrote:
What is the remainder when \(5^{91}\) is divided by 91?

1) 5
2) 8
3) 13
4) 34
5) 47



I doubt such a question will come in actuals..

Otherwise..
1) as per remainder theorem
\(91=7*13\) so LCM of (7-1,13-1) or (6,12) is 12
So \(5^{91}\) will give the same remainder as \(5^7\), since 91/12 gives 7 as remainder.
\(5^7=(5^3)^2*5=(125)^2*5\)
125 will leave 125-91=34 as remainder
So \(34*34*5=34*170\)
170 will leave a remainder of -12
\(34*(-12)= 68*(-6)=-23*-6=138\)
So remainder=138-91=47

2) if you do not know the theorem you will have to get the terms closer to multiple of 91
So \(5^{91}=(5^4)^{22}*5^3=625^{22}*5^3\)
91*7=637, so remainder is \(625-637=(-12)\)
So new term is \((-12)^{22}*5^3\)
As even power, it will be same as \(12^{22}*5^3=4^{22}*3^{22}*5^3\)

Also \(4^3=64\) and remainder is 64-91=-27
So \((4^3)^7*4*3^{22}*5^3=(-27)^7*3^{22}*5^3=(-3^3)^7*3^{22}*5^3=-(3^{43}*5^3*4)\)
Now \(3^6=729\) and 91*8=728 so remainder is 1
\(-(3^6)^7*3*5^3*4=-(1^7*3*5^3*4)=-1*3*5*100=-1*3*5*9=-135\)
So POSITIVE remainder will be \(91*2-135=182-135=47\)

E

you could have used this too to simplify
Now \(2^7*5=128*5=640\) remainder will be 640-637=3



Can you please share some link for examples on remainder theorem used in 1
User avatar
B
Sanyami19
Intern
Intern
Joined: 02 Oct 2021
Posts: 13
Own Kudos [?]: 1 [0]
Given Kudos: 163
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   12 Jul 2022, 10:44
rahulkashyap wrote:
91 = 7*13

Let us find out Rem[5^91/7] and Rem[5^91/13] - using binomial expansion
We will combine them later.

Rem [5^91/7]

= 5(5)^90 / 7
= 5( 125) ^ 30 / 7
=5 (126-1)^30 / 7

remainder = 5. (-1)^30 = 5 / 7
rem = 5



Rem [5^91/13]
= Rem 5. (5)^90 / 13
= 5. (25)^45 / 13
= 5. (26-1)^45)/ 13
remainder = 5. (-1)^45 / 13
= -5/13 = r= 8



So, our answer is a number which leaves a remainder of 5 when divided by 7 and it should leave a remainder of 8 when divided by 13.

simplifying this, we get N= 91K + 47
Hence 47 is the reaminder


How did you arrive at the equation N= 91K + 47 from this statement - is a number which leaves a remainder of 5 when divided by 7 and it should leave a remainder of 8 when divided by 13. Could you pls explain in detail?
User avatar
B
ABHISEKGMAT123
Intern
Intern
Joined: 19 Dec 2018
Posts: 16
Own Kudos [?]: 2 [0]
Given Kudos: 13
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   10 Sep 2022, 03:23
Re(5^91/ 91)
Split 91, the divisor to 13 and 7

Separately find out out the remainders with 13 and 7.
With 7,
E(7)=6, therefore 5^91 actually reduces to Rem(5^1/ 7)----------Remainder is 5; the no is of the form 7a+5
Similarly with 13,
E913)=12, therefore 5^91 actually reduces to Rem(5^7/ 7)----------Remainder is 8; the no is of the form 13b+8
Congruence of these two number types gives 47 the answer. (7a+5=13b+8)
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32679
Own Kudos [?]: 822 [0]
Given Kudos: 0
Send PM
Re: What is the remainder when 5^91 is divided by 91? [#permalink] New post   15 Dec 2023, 04:09
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 Club Bot
gmatclubot
Re: What is the remainder when 5^91 is divided by 91? [#permalink]
15 Dec 2023, 04:09
Moderators:
Bunuel
Math Expert
92915 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions