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What will be the remainder when 13^7+14^7+15^7+16^7 is div..

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What will be the remainder when 13^7+14^7+15^7+16^7 is div.. [#permalink] New post 06 Sep 2013, 12:10
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Difficulty:

  55% (hard)

Question Stats:

56% (01:43) correct 44% (01:32) wrong based on 39 sessions
What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided by 58?

a) 57
b) 1
c) 30
d) 0
e) 28


[Reveal] Spoiler:
Not quite sure how to go about this problem - does the fact that 13 + 14 + 15 + 16 add up to 58 have anything to do with the solution?
[Reveal] Spoiler: OA
Manager
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Joined: 04 Apr 2013
Posts: 153
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Kudos [?]: 26 [0], given: 36

Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div.. [#permalink] New post 06 Sep 2013, 16:25
salsal wrote:
What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided by 58?

a) 57
b) 1
c) 30
d) 0
e) 28


[Reveal] Spoiler:
Not quite sure how to go about this problem - does the fact that 13 + 14 + 15 + 16 add up to 58 have anything to do with the solution?


salsal,

probably this is not the appropriate way (though correct) to solve this problem, but it helped me to solve this under 20 sec :)


a^n + b^n + c^n will always be divisible by a+b+c is n is odd. ex:- a^3 + b^3 = (a + b) (a^2 + b^2 - ab)

a^n - b^n will be divisible by a+b if n is even. ex: - a^2 - b^2 = (a-b)(a+b)

Experts might provide you with a better way to solve this problem.
_________________

Maadhu

MGMAT1 - 540 ( Trying to improve )

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Joined: 06 Sep 2013
Posts: 1669
Location: United States
Concentration: Finance
GMAT 1: 710 Q48 V39
WE: Corporate Finance (Investment Banking)
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Kudos [?]: 168 [0], given: 276

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Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div.. [#permalink] New post 02 Dec 2013, 07:59
salsal wrote:
What will be the remainder when 13^7 + 14^7 + 15^7 + 16^7 is divided by 58?

a) 57
b) 1
c) 30
d) 0
e) 28


[Reveal] Spoiler:
Not quite sure how to go about this problem - does the fact that 13 + 14 + 15 + 16 add up to 58 have anything to do with the solution?


a^n - b^n is always divisible by a-b
a^n-b^n is always divisible by a+b whenever 'n' is even
a^n + b^n is always divisible by a+b whenever 'n' is odd and not divisible by a+b if 'n' is even

Therefore D is the correct answer
Hope this helps
Cheers
J

Last edited by jlgdr on 17 Feb 2014, 06:36, edited 1 time in total.
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Posts: 135
Location: United States
GMAT 1: 760 Q50 V42
GPA: 3.3
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Kudos [?]: 30 [0], given: 17

Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div.. [#permalink] New post 02 Dec 2013, 09:38
Tough problem.

If you're unable to determine that the sum of the exponent bases is a factor (ie, a+b+c+d is a factor), you can at least bring the odds to your favor.

The equation can also be expressed as odd + even + odd + even = even, so we know that the remainder should be even. This eliminates two of the choices, and gives you much better odds.
Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div..   [#permalink] 02 Dec 2013, 09:38
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What will be the remainder when 13^7+14^7+15^7+16^7 is div..

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