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#1 2008-11-29 06:27:21

tony123
Member
Registered: 2007-08-03
Posts: 189

the remainder

What is the remainder when 

6^83 + 8^83 is divided by 49?

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#2 2008-11-30 03:06:33

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: the remainder


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#3 2008-11-30 16:45:44

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,534

Re: the remainder

I thought the question was whether 6^83+8^83 is divisible by 14.
And the first thing I wanted to do this morning is say it is not, because 6^83+8^83 is even and 49 is odd smile

JaneFairfax' reasoning and working is indeed elegant!


Character is who you are when no one is looking.

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#4 2009-05-18 23:07:37

Fruityloop
Member
Registered: 2009-05-18
Posts: 120

Re: the remainder

JaneFairfax,
     How did you go from


to

Thank You.
Fruityloop.

Last edited by Fruityloop (2009-05-18 23:08:36)


The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

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#5 2009-05-19 00:09:58

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: the remainder


I took out the first odd power of 7 (which is 83×7). The remainder is a sum of terms involving
– which is equal to 7 times the sum of terms involving

Last edited by JaneFairfax (2009-05-19 00:13:09)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#6 2009-05-19 01:15:15

Fruityloop
Member
Registered: 2009-05-18
Posts: 120

Re: the remainder

Ok.  I was trying to remove the highest power of 7 in the binomial expansion, (7^83).  Instead of 7*83 which is the lowest odd power of 7 remaining.  In the answer you don't have to worry about the second part of the sum because it is a multiple of 49.  Very good.  I think I finally understand.
Fruityloop.


The eclipses from Algol (an eclipsing binary star) come further apart in time when the Earth is moving away from Algol and closer together in time when the Earth is moving towards Algol, thereby proving that the speed of light is variable and that Einstein's Special Theory of Relativity is wrong.

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#7 2009-10-19 20:01:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,247

Re: the remainder

Hi;

In this case there is a simple way. Just  make use of the simplest properties of modular arithmetic. We want to solve:

We are looking for modulo 1 or 48.

We are done because 6*6^6 = 6^7 which is 48 mod 49. What is so special about 48?

So the answer is 35.

The explanation takes longer than doing one of these.

Last edited by bobbym (2009-10-19 20:04:52)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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