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

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Intern
Joined: 28 Jan 2011
Posts: 21
What will be the remainder when 13^7+14^7+15^7+16^7 is div..  [#permalink]

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06 Sep 2013, 13:10
1
1
00:00

Difficulty:

45% (medium)

Question Stats:

52% (01:14) correct 48% (02:54) wrong based on 48 sessions

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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

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?
Manager
Joined: 04 Apr 2013
Posts: 111
Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div..  [#permalink]

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06 Sep 2013, 17:25
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

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.
SVP
Joined: 06 Sep 2013
Posts: 1544
Concentration: Finance
Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div..  [#permalink]

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Updated on: 17 Feb 2014, 07:36
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

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

Originally posted by jlgdr on 02 Dec 2013, 08:59.
Last edited by jlgdr on 17 Feb 2014, 07:36, edited 1 time in total.
Manager
Joined: 04 Sep 2012
Posts: 139
Location: United States
GMAT 1: 760 Q50 V42
GPA: 3.3
Re: What will be the remainder when 13^7+14^7+15^7+16^7 is div..  [#permalink]

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02 Dec 2013, 10:38
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, 10:38
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