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

tony123
Member
Registered: 2007-08-03
Posts: 228

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

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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,956

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!


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

Fruityloop
Member
Registered: 2009-05-18
Posts: 143

Re: the remainder

JaneFairfax,
     How did you go from


to

Thank You.
Fruityloop.

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

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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)

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

Fruityloop
Member
Registered: 2009-05-18
Posts: 143

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.

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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