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

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

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New post Updated on: 09 Nov 2018, 23:19
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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, 21:50.
Last edited by pushpitkc on 09 Nov 2018, 23:19, edited 1 time in total.
Formatted question, added options
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 10 Nov 2018, 04:34
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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.

Answer (A)

Check this link for more: https://www.veritasprep.com/blog/2011/0 ... ek-in-you/
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 09 Nov 2018, 21:59
Hi Leonaann
Please post the answer choices.
Moving your question to the PS forum.
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What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 09 Nov 2018, 23:37
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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)
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 10 Nov 2018, 01:13
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
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 10 Nov 2018, 01:29
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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!

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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 10 Nov 2018, 05:02
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

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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 10 Nov 2018, 05:06
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
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 10 Nov 2018, 05:11
Oh god.. i completely misread the question

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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 16 Nov 2018, 09:28
VeritasKarishma wrote:
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.

Answer (A)

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


Hi! VeritasKarishma Thanks for this explanation and sharing the link to your blog. Could you please share OG questions testing this concept.
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 18 Nov 2018, 06:42
VeritasKarishma

Hi..quick question about the signs

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

BUT we have that negative sign in front of the 2.

do we ignore the sign?

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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 18 Nov 2018, 09:55
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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 :lol:

is it valid approach ? :)
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 18 Nov 2018, 11:03
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 :lol:

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
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 02:26
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 :lol:

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 :)
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 02:31
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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,
Gladi

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 :lol:

is it valid approach ? :)

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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 05:49
can someone provide me with the link … knowledge link....how to solve similar questions ? :)
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 06:21
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

https://www.youtube.com/watch?v=trarHZy ... 4uzBuuQNST
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 06:41
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

https://www.youtube.com/watch?v=trarHZy ... 4uzBuuQNST



Mansoor50 many thanks! :) but i am not sure if that youtube guy speaks in English, does he ? :lol: the only words i understood "25", 'remainder" "-1" :lol:

have a great day :)
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What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 06:59
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 \(\frac{(63 + 1)^{10}}{21}=1\)
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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New post 19 Nov 2018, 17:19
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

https://www.youtube.com/watch?v=trarHZy ... 4uzBuuQNST



Mansoor50 many thanks! :) but i am not sure if that youtube guy speaks in English, does he ? :lol: the only words i understood "25", 'remainder" "-1" :lol:

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
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Re: What is the remainder when 61^60 is divided by 21? &nbs [#permalink] 19 Nov 2018, 17:19

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