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What is the remainder obtained when 63^26 is divided by 16?

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What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 05:10
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What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 06:35
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10



Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 06:42
chetan2u wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10



Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C



Hi Chetan sir,

What's the concept of negative remainder?Why don't we assume remainder as 15

and then (63/16)*(63^25)

Bold leaves a remainder of 15 and then no reminader.

I have a concept gap here.Please help.
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 06:59
(63^26) /16=(16*4-1)^26/16=(-1)^26=1

So choice C
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What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 07:04
prabsahi wrote:
chetan2u wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10



Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C



Hi Chetan sir,

What's the concept of negative remainder?Why don't we assume remainder as 15

and then (63/16)*(63^25)

Bold leaves a remainder of 15 and then no reminader.

I have a concept gap here.Please help.


Dear prabsahi

\(\frac{63}{16} = 15\)

And, \(\frac{63^{26}}{16} = \frac{63}{16}*\frac{63}{16}*\frac{63}{16}*\frac{63}{16}*\frac{63}{16}*\)..........26 Times...

We know that \(\frac{63}{16} = 15\)

So, \(\frac{63}{16}*\frac{63}{16} = \frac{225}{16}\) = Remainder \(1\)

We may save \(\frac{63}{16}*\frac{63}{16}\) will always yeild 1 as remainder and we have 13 such sets of \(\frac{63}{16}*\frac{63}{16}\), thus, the result will be 1 , Answer must be (C) 1

Hope this helps.
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 07:19
prabsahi wrote:
chetan2u wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10



Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C



Hi Chetan sir,

What's the concept of negative remainder?Why don't we assume remainder as 15

and then (63/16)*(63^25)

Bold leaves a remainder of 15 and then no reminader.

I have a concept gap here.Please help.



Reason, is that you have to get into big calculations then ... say with -1 as remainder we had ODD power 25 then the remainder would be -1, and remainder cannot be negative hence 16-1=15
We are taking negative remainder in initial calculations just to ease our calculations. Otherwise finally it has to be converted into positive remainder.
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 07:31
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We can treat it this way

(64-1)^26

64^26 and some lengthy calculations in the end we have 1^26. everything in between 64^26 and 1^26 will be divisible by 16

Everything in this equation is divisible by 16 except for 1 so the remainder is 1

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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 07:51
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10


For the concept behind solving such questions, check:
https://www.veritasprep.com/blog/2011/0 ... ek-in-you/

Also for negative remainders, see:
https://www.veritasprep.com/blog/2014/0 ... -the-gmat/
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 08:44
1
VeritasKarishma wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10


For the concept behind solving such questions, check:
https://www.veritasprep.com/blog/2011/0 ... ek-in-you/

Also for negative remainders, see:
https://www.veritasprep.com/blog/2014/0 ... -the-gmat/


Both the links are a gem! I still have them bookmarked: these helped me immensely during my preparation phase!
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 09:17
chetan2u wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10



Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C


I am not able to understand another point..

I dont see 16 repeated in the denominator.Its occuring only once.
say.. (63/16)*63^25

so first term can give say -1 as remainder later on its just 63^25 times..
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 09:19
VeritasKarishma wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10


For the concept behind solving such questions, check:
https://www.veritasprep.com/blog/2011/0 ... ek-in-you/

Also for negative remainders, see:
https://www.veritasprep.com/blog/2014/0 ... -the-gmat/



Many many many Thanks :)
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 09:20
enochjason wrote:
(63^26) /16=(16*4-1)^26/16=(-1)^26=1

So choice C


Also, another approach:
A mod B ( the remainder of A divided by B)
63^25 mod 16
=(16*3+15)^25 mod 16
=15^25 mod 16
=15^(2*12+1) mod 16
=(225^12) * 15 mod 16
=[(16*14+1)^12] *15 mod 16
=(1^12) *15 mod 16
=1*15 mod 16
=15
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 09:23
prabsahi wrote:
chetan2u wrote:
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10



Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C


I am not able to understand another point..

I dont see 16 repeated in the denominator.Its occuring only once.
say.. (63/16)*63^25

so first term can give say -1 as remainder later on its just 63^25 times..


You multiply the remainders ...
Say a number is 3*5 so when divided by 3 remainders will become 0*2=0
But if number is 4*5.. remainder will become 1*2=2 check 4*5=20and 20 divided by 3 gives a remainder of 2

So you have to divide each term in numerator by the same denominator ...
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 09:28
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10[/quote]


Hi....
63 divided by 16 will leave a remainder of -1 as 64 is divisible by 16..
So 63*63*63....26times will leave a remainder of (-1)*(-1)*.....26times = \((-1)^{26}=((-1)^2)^{13}=1^{13}=1\)

C[/quote]

I am not able to understand another point..

I dont see 16 repeated in the denominator.Its occuring only once.
say.. (63/16)*63^25

so first term can give say -1 as remainder later on its just 63^25 times..[/quote]

You multiply the remainders ...
Say a number is 3*5 so when divided by 3 remainders will become 0*2=0
But if number is 4*5.. remainder will become 1*2=2 check 4*5=20and 20 divided by 3 gives a remainder of 2

So you have to divide each term in numerator by the same denominator ...[/quote]



Perfect..This was really very helpful.

Many many many Thanks !!
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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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New post 29 Oct 2018, 16:54
rencsee wrote:
What is the remainder obtained when \(63^{26}\) is divided by 16?

A. -2
B. -1
C. 1
D. 8
E. 10


This can be written as -1^26 because 63 leaves 15 or -1 remainder with 16.

-1^even is 1.

Answer is 1. C.
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Re: What is the remainder obtained when 63^26 is divided by 16? &nbs [#permalink] 29 Oct 2018, 16:54
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