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

It is currently 19 Nov 2019, 00:25

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

What is the remainder when (25^99 x 4^99)^99 is divided by 11?

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

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 59126
What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:00
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

69% (01:16) correct 31% (01:27) wrong based on 294 sessions

HideShow timer Statistics

Most Helpful Community Reply
Manager
Manager
User avatar
G
Joined: 08 Jan 2018
Posts: 129
CAT Tests
What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post Updated on: 20 Jul 2019, 01:51
3
3
The given number is very big. It has to be in some pattern.

If we take n = 99 then the number is given in the form:
\((25^n * 4^n)^n\)

Let us find the value with lower terms of n-
n = 1
(25*4) = 100
Remainder (\(\frac{100}{11}\)) = 1

n = 2
\((25^2 * 4^2)^2\) = \((625 * 16)^2\) = \((10000)^2\)
Remainder (\(\frac{10000^2}{11}\)) = 1

So, for any value of n, the remainder of \((25^n * 4^n)^n\) divided by 11 will always be 1.

Answer A

Originally posted by Sayon on 19 Jul 2019, 08:10.
Last edited by Sayon on 20 Jul 2019, 01:51, edited 2 times in total.
General Discussion
Director
Director
User avatar
P
Joined: 16 Jan 2019
Posts: 500
Location: India
Concentration: General Management
WE: Sales (Other)
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:09
3
\((25^{99}∗4^{99})^{99} = (100)^{99*99}\)

100 leaves a remainder of 1 when divided by 11 and so (100)^{99*99} must leave a remainder of 1 when divided by 11
Manager
Manager
avatar
G
Joined: 29 Nov 2018
Posts: 148
Location: India
Concentration: Entrepreneurship, General Management
GPA: 3.99
WE: Engineering (Computer Hardware)
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:12
1
What is the remainder when (25^99∗4^99)^99 is divided by 11?

The product is the question stem can be written as (100^99)^99. Now any power of 100 if divided by 99 the reminder is going to be 1.
For eg 100^1 divided by 99. Remainder is 1
100^2 = 10000 divided by 99. Reminder is 1.

Hence Answer = A
Senior Manager
Senior Manager
avatar
P
Joined: 27 Aug 2014
Posts: 358
Location: Netherlands
Concentration: Finance, Strategy
Schools: LBS '22, ISB '21
GPA: 3.9
WE: Analyst (Energy and Utilities)
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:12
1
IMO answer is A:

the statement can be written as 100^(99*99). As 1 followed by any number of zeros and divided by 11, will give a reminder of 1.
so A
SVP
SVP
User avatar
P
Joined: 03 Jun 2019
Posts: 1849
Location: India
Premium Member Reviews Badge CAT Tests
What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post Updated on: 19 Jul 2019, 10:12
1
What is the remainder when (25^99∗4^99)^99 is divided by 11?

A. 1
B. 3
C. 7
D. 9
E. 10

Binomial expansion of
\((x+y)^n = x^n + C_1^n x^n-1 y + .... C_k^n x^n-k y^k +.....y^n\)

(25^99∗4^99)^99 = (100^99)^99 = ((9*11+1)^99)^99
(9*11+1)^99 = multiples of 11 + 1^99 = 11k+1
((9*11+1)^99)^99 = (11k+1)^99 = multiples of 11 + 1^99 = 11y+1

=> Remainder when (25^99∗4^99)^99 is divided by 11 = 1

IMO A
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com

Originally posted by Kinshook on 19 Jul 2019, 08:22.
Last edited by Kinshook on 19 Jul 2019, 10:12, edited 2 times in total.
Manager
Manager
avatar
G
Joined: 28 Feb 2014
Posts: 180
Location: India
Concentration: General Management, International Business
GPA: 3.97
WE: Engineering (Education)
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:25
1
We know that a^n * b^n = (ab)^6
(25^99 ∗ 4^99)^99 = (100^99)^99 = 100^(99*99)

When 100 is divided by 11, remainder is 1
similarly when 100^(99*99) is divided by 11, remainder is 1^(99*99) = 1

A is correct.
Manager
Manager
User avatar
G
Joined: 10 Mar 2019
Posts: 75
Location: Russian Federation
Schools: Booth '22, Haas '22
GMAT 1: 730 Q50 V39
GPA: 3.95
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:25
1
1. Multiply 25^99 x 4^99=100^99=10^100
2. Now we have to multiply powers and we get 10^9900
3. Now, use the binomial theorem (11-1)^9900.
4. Since the power is even then we have positive remainder 1 and this is the answer.


IMO A
Senior Manager
Senior Manager
User avatar
G
Joined: 05 Mar 2017
Posts: 261
Location: India
Concentration: General Management, Marketing
GPA: 3.6
WE: Marketing (Hospitality and Tourism)
Reviews Badge CAT Tests
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:25
What is the remainder when (2599∗499)99 is divided by 11?

The questions test your ability with regards to divisibility.

We know that units digit of 5 is always 5 and 4's unit digit since 99 is odd is 4 now (5*4) ^11 = 20^99 / 11 = 20/11
which is 9.


A. 1
B. 3
C. 7
D. 9
E. 10

hence the correct answer is 9. D.
Manager
Manager
avatar
S
Joined: 12 Apr 2017
Posts: 142
Location: United States
Concentration: Finance, Operations
GPA: 3.1
CAT Tests
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:28
1
What is the remainder when (25^99∗4^99)^99is divided by 11?

Same exponent so can combine 100^99 is going to end in 0, which that result raised to 99 will end in a 0.

some large base 10 number will end with remainder of 1.

Ex: 100/11 = 9 R 1
Director
Director
avatar
G
Joined: 22 Nov 2018
Posts: 562
Location: India
GMAT 1: 640 Q45 V35
GMAT 2: 660 Q48 V33
GMAT ToolKit User Premium Member
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:29
1
(25^99∗4^99)^99 can be written in the form (100^99)^99 as (25*25**25....99 times)*(4*4*4...99 times) will give 100*100*100..99 times

100/11 will have a remainder of 1. So (1^99)^99/11 will still be 1 (Using remainder theorem for polynomials)

IMO A - 1
_________________
Give +1 kudos if this answer helps..!!
Manager
Manager
avatar
G
Joined: 26 Jan 2016
Posts: 181
CAT Tests
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:35
1
(25^99∗4^99)^99
Units digit 5 to any power is 5
Units digit 4 to odd power is 4 & even power is 6.In this case unit digit will be 4
Now,unit digit 5*4=20
So, we're looking for solution for units digit 0^99

Multiple of 10 divided by 11 will always leave remainder as 1.

Hence A

Posted from my mobile device
_________________
Your Kudos can boost my morale..!!

I am on a journey. Gradually I'll there..!!
Manager
Manager
avatar
S
Joined: 08 Jan 2018
Posts: 98
Location: India
GPA: 4
WE: Information Technology (Computer Software)
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:36
1
What is the remainder when (25^{99}∗4^{99})^{99} is divided by 11?
\(\frac{(25^{99}∗4^{99})^{99}}{11}\)
=> \(\frac{((25*4)^{99})^{99}}{11}\)
=> \(\frac{((100)^{99})^{99}}{11}\)
=> \(\frac{((1)^{99})^{99}}{11}\) (Using Remainder Theorem or dividing 100 by 11: Remainder is 1)
=> \(\frac{(1)^{99}}{11}\)
=> \(\frac{1}{11}\)
=> 1

IMO the answer is A.

Please hit kudos if you like the solution.
Director
Director
User avatar
V
Joined: 28 Jul 2016
Posts: 652
Location: India
Concentration: Finance, Human Resources
GPA: 3.97
WE: Project Management (Investment Banking)
Reviews Badge
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:36
1
\((25^{99}∗4^{99})^{99}\) is divided by 11
simplify it to
\((100^{99})^{99}\)
now 100 when divided by 11 leaves a remainder 1
\(100 = 9*11 +1\)
thus remainder is \((1^{99})^{99}/11\) =1
Hence A
Director
Director
User avatar
P
Joined: 04 Sep 2015
Posts: 662
Location: India
WE: Information Technology (Computer Software)
Premium Member Reviews Badge
What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post Updated on: 19 Jul 2019, 08:45
1
IMO : A

What is the remainder when (25^99∗4^99)^99 is divided by 11?

A. 1
B. 3
C. 7
D. 9
E. 10


SOL:

(a^x)(b^x)=(ab)^x so we get (100^99)99.

Now if the power of 100 is odd then the last multiple of 11 before that number will be 999... or (100-1)

so this means we will always have remainder as 1.

Originally posted by abhishekdadarwal2009 on 19 Jul 2019, 08:36.
Last edited by abhishekdadarwal2009 on 19 Jul 2019, 08:45, edited 1 time in total.
Manager
Manager
avatar
G
Joined: 30 May 2018
Posts: 157
Location: Canada
GMAT 1: 710 Q49 V36
GPA: 3.8
Premium Member Reviews Badge
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:37
1
A

The given expression can be written as (10)^2*99*99. Now, notice that when 10^n is divided by 11, remainder is 10 when is odd and 1 when n is even.

Example: 10/11 - remainder is 10
100/11 - remainder is 1
1000/11 - remainder is 10
10000/11 - remainder is 1

So, in the given expression, n = 2*99*99 - even no. So remainder should be 1.
Manager
Manager
avatar
S
Joined: 18 Feb 2017
Posts: 93
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:39
1
IMO A.

\((25^{99}∗4^{99})^{99} --> (25∗4)^{99 * 99}\)
25*4= 100.
When 100 is divided by 11 we get a remainder 1.
This is equivalent to having \(\frac{(25∗4)^{99 * 99}}{11}\) giving us,

\(\frac{(1)^{99 * 99}}{11}\) --> Which is 1/11 , giving a remainder of 1.
Senior Manager
Senior Manager
User avatar
P
Joined: 10 Jan 2017
Posts: 329
Location: India
Reviews Badge CAT Tests
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:39
1
IMO the correct answer is option A - 1
Explanation given as attachment -
Attachments

IMG_20190719_210529.JPG
IMG_20190719_210529.JPG [ 1017.33 KiB | Viewed 2052 times ]


_________________
Good, better, best. Never let it rest. 'Till your good is better and your better is best.
Please hit +1 Kudos if you like my Post.
Director
Director
avatar
P
Joined: 24 Nov 2016
Posts: 783
Location: United States
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:45
Quote:
What is the remainder when (25ˆ99∗4ˆ99)ˆ99 is divided by 11?

A. 1
B. 3
C. 7
D. 9
E. 10


(25ˆ99∗4ˆ99)ˆ99 = [5ˆ(2*99)•2ˆ(2*99)]ˆ99 = [10ˆ198]ˆ99 = [(11-1)ˆ198]ˆ99
now, 11 to any power is divisible by 11, so what we need to find is the remainder of [(-1)ˆ198]ˆ99
which is the same as finding the remainder of -1/11
the remainder of a negative is the same as subtracting 11-1=10

Answer (E).
Manager
Manager
avatar
S
Joined: 30 Aug 2018
Posts: 103
Location: India
Concentration: Finance, Accounting
GPA: 3.36
WE: Consulting (Computer Software)
Premium Member
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?  [#permalink]

Show Tags

New post 19 Jul 2019, 08:45
1
Simplify.
100^(99*99)
100/11 remainder is 1
1^(99*99)
Therefore, answer=1
GMAT Club Bot
Re: What is the remainder when (25^99 x 4^99)^99 is divided by 11?   [#permalink] 19 Jul 2019, 08:45

Go to page    1   2   3   4   5    Next  [ 88 posts ] 

Display posts from previous: Sort by

What is the remainder when (25^99 x 4^99)^99 is divided by 11?

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





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne