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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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Updated on: 09 Nov 2018, 23:19
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Difficulty:

65% (hard)

Question Stats:

53% (01:14) correct 47% (01:35) wrong based on 163 sessions

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

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10 Nov 2018, 04:34
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/
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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09 Nov 2018, 21:59
Hi Leonaann
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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09 Nov 2018, 23:37
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)
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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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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10 Nov 2018, 01:29
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!

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

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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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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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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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16 Nov 2018, 09:28
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/

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

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18 Nov 2018, 06:42

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

do we ignore the sign?

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

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18 Nov 2018, 09:55
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 ?
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Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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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

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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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

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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19 Nov 2018, 02:31
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 ?

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Posts: 1287
Re: What is the remainder when 61^60 is divided by 21?  [#permalink]

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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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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

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

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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

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

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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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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

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? &nbs [#permalink] 19 Nov 2018, 17:19

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