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What is the remainder when
6^83 + 8^83 is divided by 49?
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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!
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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JaneFairfax,
How did you go from
Last edited by Fruityloop (2009-05-18 23:08:36)
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Last edited by JaneFairfax (2009-05-19 00:13:09)
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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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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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