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

What is the remainder when

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

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,730

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 is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 135

JaneFairfax,

How did you go from

to

Thank You.

Fruityloop.

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

**The eclipses from Algol 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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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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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**Fruityloop****Member**- Registered: 2009-05-18
- Posts: 135

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 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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,389

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.****No great discovery was ever made without a bold guess. **

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