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What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b

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Bunuel
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What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink] New post   06 Oct 2017, 04:49
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What will be the remainder when \(13^7 + 14^7 + 15^7 + 16^7\) is divided by 58?

A. 0
B. 1
C. 28
D. 30
E. 57
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chetan2u
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Re: What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink] New post   06 Oct 2017, 06:41
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Bunuel wrote:

CHALLENGE QUESTIONS



What will be the remainder when \(13^7 + 14^7 + 15^7 + 16^7\) is divided by 58?

A. 0
B. 1
C. 28
D. 30
E. 57


hi..

Here we have to remember that \(a^n+b^n\) is always div by a+b when n is ODD

so 58 should tell you that 58=2*29 and the numbers in equation also add up to 29 - 13+16 and 14+15
\(13^7 + 14^7 + 15^7 + 16^7= (13^7+16^7)+(14^7+15^7)\)

two things
1) the entire sum is \(Odd^7+Even^7+Odd^7+Even^7\).. here total will be EVEN
2) now \(13^7+16^7\) will be div by 13+16=29 and similarly \(14^7+15^7.\).

so the the EQUATION is div by 2 and also by 29..

thus remainder will be 0

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Pallabi89
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Re: What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink] New post   01 Jan 2018, 19:56
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The numbers can be added as = (13+14+15+16)^7 =(58)^7 {followed x^n+y^n=(x+y)^n}

Above divided by 58 would give u reminder 0.

Plz let me know in case this approach is correct.
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Re: What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink] New post   01 Jan 2018, 20:24
Pallabi89 wrote:
The numbers can be added as = (13+14+15+16)^7 =(58)^7 {followed x^n+y^n=(x+y)^n}

Above divided by 58 would give u reminder 0.

Plz let me know in case this approach is correct.



x^n + y^n is not equal to (x+y)^n
e.g.
(x+y)^2 = (x+y) X (x+y) = x^2+ y^2 + xy +xy = x^2 + y^2 + 2xy
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Re: What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink] New post   01 Jan 2018, 21:16
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Pallabi89 wrote:
The numbers can be added as = (13+14+15+16)^7 =(58)^7 {followed x^n+y^n=(x+y)^n}

Above divided by 58 would give u reminder 0.

Plz let me know in case this approach is correct.



Hi...

You cannot take a case in isolation and generalized..

Firstly x^n+y^n is not equal to (x+y)^n unless n is 1 or one of x andy is 0..
There may some more cases here.

Now incase you meant for checking divisibility ONLY..
Check whether n is odd or even..
If odd x^n+y^n is div by x+y otherwise not..

Example 2^2+4^2 is not div by 2+4 or 6

So be careful
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Re: What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink] New post   10 Mar 2024, 18:49
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Re: What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided b [#permalink]
10 Mar 2024, 18:49
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