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

 It is currently 17 Jul 2019, 04:21 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # What is the remainder when 61^60 is divided by 21?

Author Message
TAGS:

### Hide Tags

Manager  B
Joined: 22 Sep 2018
Posts: 65
What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
13 00:00

Difficulty:   55% (hard)

Question Stats: 53% (01:50) correct 47% (02:05) wrong based on 183 sessions

### HideShow timer Statistics What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

Originally posted by Leonaann on 09 Nov 2018, 22:50.
Last edited by pushpitkc on 10 Nov 2018, 00:19, edited 1 time in total.
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9442
Location: Pune, India
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

5
1
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

Use Binomial theorem twice on it.

$$(61)^{60} = (63 - 2)^{60}$$

When this is divided by 21, all terms with 63 will be completely divisible by 21 and remainder will be the last term $$2^{60}$$

Now, we need to focus on this.

2^{60} = (2^6)^{10} = 64^{10} = (63 + 1)^{10}

When this is divided by 21, all terms with 63 will be completely divisible by 21 and remainder will be the last term 1.

Check this link for more: https://www.veritasprep.com/blog/2011/0 ... ek-in-you/
_________________
Karishma
Veritas Prep GMAT Instructor

##### General Discussion V
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 1028
Location: India
GPA: 3.64
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

Hi Leonaann
Moving your question to the PS forum.
_________________
Please give kudos, if you like my post

When the going gets tough, the tough gets going...
Senior PS Moderator V
Joined: 26 Feb 2016
Posts: 3360
Location: India
GPA: 3.12
What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

2
1
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

We can write $$61^{60}$$ as $$(42+19)^{60}$$

Since 42 is completely divisible by 21, we need to work on $$19^{60}$$ only. $$19^{60}$$ can be written as $$(21 - 2)^{60}$$.

The remainder when the expression is divided by 21 is $$(-2)^{10}$$ or $$(64)^{10}$$.

Now, $$64^{10}$$ can be written as $$(63 + 1)^{10}$$, making the remainder of the expression when divided by 21 -> $$1^{10} = 1$$

Therefore, the remainder when 61^60 is divided by 21 is 1(Option A)
_________________
You've got what it takes, but it will take everything you've got
Manager  B
Joined: 28 Jun 2018
Posts: 74
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

Hello,

I split it like this (63-2)^60
Now dealing with 2^60
I don't really know how to determine the remainder from this. Can someone help? pushpitkc
Director  G
Joined: 19 Oct 2013
Posts: 525
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
1
hibobotamuss wrote:
Hello,

I split it like this (63-2)^60
Now dealing with 2^60
I don't really know how to determine the remainder from this. Can someone help? pushpitkc

Hi 2^6 = 64 so the 2^60 becomes 64^10

64 is (63+1)^10 so we have 63 that is divisible by 21 and 1^10

Hope this helps!

Posted from my mobile device
Manager  G
Joined: 10 Jan 2013
Posts: 243
Location: India
Concentration: General Management, Strategy
GPA: 3.95
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
61 can be written as (60+1)^60

While 60 is always divisible by 2(i.e. remainder is Zero)

1^60 will always leave 1 as remainder.

So A

Posted from my mobile device
Senior PS Moderator V
Joined: 26 Feb 2016
Posts: 3360
Location: India
GPA: 3.12
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

saurabh9gupta wrote:
While 60 is always divisible by 2(i.e. remainder is Zero)

saurabh9gupta - in this question, you are asked to calculate the remainder when 61^60 is divided by 21
_________________
You've got what it takes, but it will take everything you've got
Manager  G
Joined: 10 Jan 2013
Posts: 243
Location: India
Concentration: General Management, Strategy
GPA: 3.95
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

Oh god.. i completely misread the question

Posted from my mobile device
Director  V
Joined: 12 Feb 2015
Posts: 875
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

Use Binomial theorem twice on it.

$$(61)^{60} = (63 - 2)^{60}$$

When this is divided by 21, all terms with 63 will be completely divisible by 21 and remainder will be the last term $$2^{60}$$

Now, we need to focus on this.

2^{60} = (2^6)^{10} = 64^{10} = (63 + 1)^{10}

When this is divided by 21, all terms with 63 will be completely divisible by 21 and remainder will be the last term 1.

Check this link for more: https://www.veritasprep.com/blog/2011/0 ... ek-in-you/

_________________
"Please hit +1 Kudos if you like this post" _________________
Manish "Only I can change my life. No one can do it for me"
Manager  B
Joined: 29 May 2017
Posts: 127
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

(63 - 2)^60/21 --> the remainder of 2^60/21 is 1.

do we ignore the sign?

regards
VP  D
Joined: 09 Mar 2016
Posts: 1273
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

i noticed that everytime when we multiply 61 by itself the unit digit is always 1, so i chose A is it valid approach ? Intern  B
Joined: 20 Oct 2017
Posts: 8
Location: India
Concentration: General Management, Strategy
GPA: 3.22
WE: Operations (Manufacturing)
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

dave13 wrote:
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

i noticed that everytime when we multiply 61 by itself the unit digit is always 1, so i chose A is it valid approach ? By that logic ... the answer would be same if the number were 71, 81, 91 or any other number with one as unit digit.
In short NOOO
VP  D
Joined: 09 Mar 2016
Posts: 1273
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

roysaurabhkr wrote:
dave13 wrote:
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

i noticed that everytime when we multiply 61 by itself the unit digit is always 1, so i chose A is it valid approach ? By that logic ... the answer would be same if the number were 71, 81, 91 or any other number with one as unit digit.
In short NOOO

Actually i didnt get this concept, maybe you can suggest how to solve similar questions thank you Senior PS Moderator D
Status: It always seems impossible until it's done.
Joined: 16 Sep 2016
Posts: 751
GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
1
Could have worked if the divisor was a one digit integer, however, since we are dividing by 21, the remainder could be anything between 0 and 20. Hence the units place of 1 can be either a remainder of 1 or 11.

Hence this approach is not valid. ( If 11 was one option then there was no way you could choose 1 over 11 or vice-versa using this method)

Best,

dave13 wrote:
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

i noticed that everytime when we multiply 61 by itself the unit digit is always 1, so i chose A is it valid approach ? _________________
Regards,

“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back)
VP  D
Joined: 09 Mar 2016
Posts: 1273
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

can someone provide me with the link … knowledge link....how to solve similar questions ? Manager  B
Joined: 29 May 2017
Posts: 127
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
dave13 wrote:
can someone provide me with the link … knowledge link....how to solve similar questions ? i found this and similar videos very helpful

VP  D
Joined: 09 Mar 2016
Posts: 1273
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

Mansoor50 wrote:
dave13 wrote:
can someone provide me with the link … knowledge link....how to solve similar questions ? i found this and similar videos very helpful

Mansoor50 many thanks! but i am not sure if that youtube guy speaks in English, does he ? the only words i understood "25", 'remainder" "-1" have a great day Senior Manager  D
Joined: 18 Jun 2018
Posts: 267
What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
Leonaann wrote:
What is the remainder when 61^60 is divided by 21?

A. 1
B. 2
C. 3
D. 4
E. 5

OA:A

Remainder $$\frac{61^{60}}{21}=$$ Remainder $$\frac{(-2)^{60}}{21}=$$ Remainder$$\frac{{(2^6)}^{10}}{21} =$$Remainder $$\frac{64^{10}}{21} =$$ Remainder $$\frac{(63 + 1)^{10}}{21}=1$$
Manager  B
Joined: 29 May 2017
Posts: 127
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

### Show Tags

1
dave13 wrote:
Mansoor50 wrote:
dave13 wrote:
can someone provide me with the link … knowledge link....how to solve similar questions ? i found this and similar videos very helpful

Mansoor50 many thanks! but i am not sure if that youtube guy speaks in English, does he ? the only words i understood "25", 'remainder" "-1" have a great day dave13

My apologies for that snafu.

if you type in "remainders gmat" in youtube you will get a ton of videos that start from the basics then go up.

regards Re: What is the remainder when 61^60 is divided by 21?   [#permalink] 19 Nov 2018, 18:19

Go to page    1   2    Next  [ 21 posts ]

Display posts from previous: Sort by

# What is the remainder when 61^60 is divided by 21?  