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

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Leonaann
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What is the remainder when 61^60 is divided by 21? [#permalink] New post   Updated on: 10 Nov 2018, 00:19
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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
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A

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.
Formatted question, added options
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KarishmaB
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What is the remainder when 61^60 is divided by 21? [#permalink] New post   Updated on: 20 Oct 2022, 05:08
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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)

Originally posted by KarishmaB on 10 Nov 2018, 05:34.
Last edited by KarishmaB on 20 Oct 2022, 05:08, edited 1 time in total.
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pushpitkc
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What is the remainder when 61^60 is divided by 21? [#permalink] New post   10 Nov 2018, 00:37
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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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hibobotamuss
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Re: What is the remainder when 61^60 is divided by 21? [#permalink] New post   10 Nov 2018, 02:13
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] New post   10 Nov 2018, 02:29
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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] New post   19 Nov 2018, 06:49
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] New post   19 Nov 2018, 07:21
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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] New post   05 Jan 2019, 13:59
Mansoor50 wrote:
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?

regards


Mansoor50
You can ignore it in this case because power is even and negative sign raised to even power becomes positive. E.g -1(^2) = -1*-1 = 1
Also in this case, even if our remainder would be -1, it would be equivalent to remainder being 1 since our divisor is 2 (I believe you'll know this concept by now) :)
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Re: What is the remainder when 61^60 is divided by 21? [#permalink] New post   01 Jul 2023, 07:46
Asked: What is the remainder when 61^60 is divided by 21?

61 = 21*3 - 2
61^60 = (21*3-2)^60

the remainder when 61^60 is divided by 21
= the remainder when (-2)^60 is divided by 21.
= the remainder when (2^6)^10 is divided by 21.
= the remainder when (21*3+1)^10 is divided by 21.
= the remainder when 1 is divided by 21.
=1

IMO A
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Re: What is the remainder when 61^60 is divided by 21? [#permalink] New post   21 Dec 2023, 14:56
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I dont really get how you come up with all these numbers and how in the end you get remainder 1...
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Re: What is the remainder when 61^60 is divided by 21? [#permalink] New post   22 Dec 2023, 00:12
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jennisan wrote:
I dont really get how you come up with all these numbers and how in the end you get remainder 1...


Hoe the links below will help.

6. Remainders



For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread
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Re: What is the remainder when 61^60 is divided by 21? [#permalink] New post   22 Dec 2023, 22:55
jennisan wrote:
I dont really get how you come up with all these numbers and how in the end you get remainder 1...


I did not even break 61 into two numbers. I did it in the following way.
1. Given 61^60. We can write it as (61^2)^30 = 3721^30. Now divide 3721 by 21, the remainder is 4. So write 4^30. We will deal with this value only now.
2. Now write 4^30 as (4^3)^10 = 64^10. Divide 64 by 21, the remainder is 1. We need this value now.
3. Finally write 1^10 which yields 1. If we divide it by 21 then the remainder will be 1 only, and this is the answer.
The intention here is to utilize the power in such a way so that when we divide it by the denominator, we get a very small value in the numerator which will be our remainder.
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Re: What is the remainder when 61^60 is divided by 21? [#permalink]
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