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

Question

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

KarishmaB · Accepted Answer

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)

pushpitkc · Answer

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)

hibobotamuss · Answer

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

Salsanousi · Answer

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

dave13 · Answer

can someone provide me with the link … knowledge link....how to solve similar questions ?

Mansoor50 · Answer

i found this and similar videos very helpful https://www.youtube.com/watch?v=trarHZy ... 4uzBuuQNST

Darshi04 · Answer

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)

Kinshook · Answer

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

jennisan · Answer

I dont really get how you come up with all these numbers and how in the end you get remainder 1...

Bunuel · Answer

Hoe the links below will help.

6. Remainders 
     Theory   
 REMAINDERS 
 Remainders: Tips and hints 
 Divisibility and Remainders on the GMAT 
 Remainders on the GMAT 
 A Remainders Shortcut for the GMAT 
 All About Negative Remainders on the GMAT 
 A Tricky Question on Negative Remainders 
 Compilation of tips and tricks to deal with remainders 
 Cyclicity and the remainders on the GMAT 
 GMAT Question Patterns: Remainders 
     Questions   
 Units digits, exponents, remainders problems  
 DS Questions 
 PS Questions

For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

samarpan.g28 · Answer

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.

bumpbot · Answer

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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?

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