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#2 2008-12-01 02:06:33
#3 2008-12-01 15:45:44
- ganesh
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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 
JaneFairfax' reasoning and working is indeed elegant!
Character is who you are when no one is looking.
#4 2009-05-19 21:07:37
- Fruityloop
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Re: the remainder
JaneFairfax,
How did you go from
to
Thank You.
Fruityloop.
Last edited by Fruityloop (2009-05-19 21:08:36)
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#5 2009-05-19 22:09:58
- JaneFairfax
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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 22:13:09)
#6 2009-05-19 23:15:15
- Fruityloop
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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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July 2012 was the hottest month ever for the U.S.
#7 2009-10-20 19:01:02
- bobbym
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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 (2009-10-20 19:04:52)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.