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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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29 Oct 2018, 06:10
1
5
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Difficulty:

35% (medium)

Question Stats:

70% (01:08) correct 30% (01:23) wrong based on 63 sessions

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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
Math Expert
Joined: 02 Aug 2009
Posts: 7984
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 07:35
1
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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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 07:42
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.

Intern
Joined: 03 Oct 2018
Posts: 4
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 07:59
(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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29 Oct 2018, 08:04
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.

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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Math Expert
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Posts: 7984
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 08:19
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.

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

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29 Oct 2018, 08:31
2
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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29 Oct 2018, 08:51
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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29 Oct 2018, 09:44
1
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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29 Oct 2018, 10:17
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..
Senior Manager
Joined: 09 Jun 2014
Posts: 352
Location: India
Concentration: General Management, Operations
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 10:19
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
Intern
Joined: 03 Oct 2018
Posts: 4
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 10:20
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
Math Expert
Joined: 02 Aug 2009
Posts: 7984
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 10:23
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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Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 10:28
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]

Many many many Thanks !!
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Joined: 20 Sep 2018
Posts: 37
Re: What is the remainder obtained when 63^26 is divided by 16?  [#permalink]

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29 Oct 2018, 17:54
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.

Re: What is the remainder obtained when 63^26 is divided by 16?   [#permalink] 29 Oct 2018, 17:54
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