Prove that is divisible by 7.

I set about trying to show that this expression is equal to .

I have that:

The question also has Fermat's little theorem above it so I think I should be using that somewhere.

Anyone have any ideas?

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- June 3rd 2009, 12:51 AMShowcase_22Divisibility by 7
Prove that is divisible by 7.

I set about trying to show that this expression is equal to .

I have that:

The question also has Fermat's little theorem above it so I think I should be using that somewhere.

Anyone have any ideas? - June 3rd 2009, 01:17 AMMoo
Hello,

Let's take for example

It's congruent to

Now, you know by Fermat's little theorem, that

As you noticed,

It can be proved that hence

Why ? Because

And then :) - June 3rd 2009, 01:30 AMTheAbstractionist
- June 3rd 2009, 01:47 AMShowcase_22Quote:

Hello,

I finally got it to work out using TheAbstractionist's method (I looked at what you did, did it for 2222 whilst looking at yours and then did it for 5555 separately).

I'll have a go using your method now Moo since it seems to be the one in the mark scheme. - June 3rd 2009, 02:08 AMShowcase_22
Okay, here's my way of doing it. I think it's both of your methods combined (or I may just be copying one of your methods, i'm not that good at number theory!):

by Fermat's little theorem

by Fermat's little theorem.

(Rofl) **WIN** (Rofl) - June 3rd 2009, 02:38 AMMoo
Well, actually that was exactly the way I wanted you to do (Rofl)

Good job.

Though your notations are very strange...

Quote:

Quote:

- June 3rd 2009, 02:40 AMShowcase_22
Ah, sorry about my notations!

I didn't realise there was a command for . The more I know!

That second one was just plain wrong, sorry I wrote it!

On the plus side:

__Spoiler__: