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# what is the remainder ?

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Intern
Joined: 23 Sep 2008
Posts: 8
what is the remainder ? [#permalink]

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23 Apr 2009, 08:57
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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

Could someone pls elaborate whats the approach for these questions ??
Manager
Joined: 15 Dec 2008
Posts: 52
Schools: HBS(08) - Ding. HBS, Stanford, Kellogg, Tuck, Stern, all dings. Yale - Withdrew App. Emory Executive -- Accepted, Matriculated, Withdrewed (yes, I spelled it wrong on purpose). ROSS -- GO BLUE 2011.
Re: what is the remainder ? [#permalink]

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23 Apr 2009, 11:19
mavesum 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

Could someone pls elaborate whats the approach for these questions ??

13+14+15+16=58

the whole mess (13^7+....) will be divisable by 58, since all of them have the same exponent.

1^x + 2^x will always be divisible by 3 for interger x>0

same thing for any combination of foolishness they give you.

remainder is 0.
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1346
Re: what is the remainder ? [#permalink]

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23 Apr 2009, 16:50
RahlowJenkins wrote:
1^x + 2^x will always be divisible by 3 for interger x>0

No, that's not true - 1^2 + 2^2 is not divisible by 3, for example. What you're saying is true when x is odd, and not true when x is even.

I responded to the question in the original post on BTG by rearranging terms and factoring, but I can give a different solution here, using mathematics you definitely don't need to know for the test. But since modular arithmetic will be familiar to some people here, I'll use it on the above question.

Here the $$\equiv$$ symbol means "has the same remainder as" and "mod 29" means "when dividing by 29". Modular arithmetic establishes that, if you're interested in finding the remainder of a sum, product, or difference, you can replace one number in that sum, product or difference with any other number that gives the same remainder).

$$13^7 + 14^7 + 15^7 + 16^7 \equiv 13^7 + 14^7 + (-14)^7 + (-13)^7 \equiv 0 \text{ mod 29}$$

So the sum is divisible by 29. Since it's even, it must also be divisible by 58.

Here I'm using the fact that 15 and -14 both have the same remainder when divided by 29, since they're both 15 more than a multiple of 29 (-14 = -29 + 15). Similarly, 16 and -13 have the same remainder when divided by 29.

You certainly don't ever need modular arithmetic on the GMAT, however, and the above question is not a realistic test question.
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Intern
Joined: 24 Mar 2009
Posts: 6
Re: what is the remainder ? [#permalink]

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23 Apr 2009, 21:23
I cannot agree that this is unrealistic question, 2 days ago I took my GMAT and I encountered with very similar problem, to which I still do not know a quick solution, it was something like this:

2+2^1+2+^2+2^3+2^4+2^5+2^6 =?

Aswers were all in similar form (but I dont remembe the precicely):

2^11
2^15
2^7
or something like that.

Can anyone help?
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1346
Re: what is the remainder ? [#permalink]

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24 Apr 2009, 02:31
yns, those are very different questions, and the mathematics needed to solve them is entirely separate. If you're referring to a question you saw on a real GMAT, you should not be posting it here. There is, however, a GMATPrep question that asks:

2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = ?

If you notice that 2 + 2 = 2^2, and 2^2 + 2^2 = 2^3, etc, you can see that

2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = 2^2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = 2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = ... = 2^9
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Senior Manager
Joined: 05 Apr 2007
Posts: 255
Re: what is the remainder ? [#permalink]

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24 Apr 2009, 11:32
Adding all the numbers, the result is 7 * 58. So the remainder, on dividing by 58 will be O.
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Re: what is the remainder ?   [#permalink] 24 Apr 2009, 11:32
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