Fermat's Little Theorem.
a^{p-1}=(corresponds to)1mod p
using this, we have remainder when 3^100 is divided by 101 as 1.
Is this correct ?
Can i have few more examples where this theorem is applied but not so directly...
@mukushla

- hartnn

- schrodinger

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- hartnn

- hartnn

yes, that was direct use of the theorem....

- anonymous

i think we just use it directly

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## More answers

- hartnn

any other interesting topic or theorem in modular arithmetic ?

- anonymous

Wilson's theorem is interesting

- hartnn

(n-1)!=-1(mod n)
how do we use it ?

- anonymous

If \(p\) is a prime, then \((p-1)!+1\) is a multiple of \(p\), that is\[(p-1)! \equiv -1 \ \ \text{mod} \ p\]

- hartnn

i will be glad if someone posts a link for a good reference in this topic...

- anonymous

FLT is also a^p = a (mod p)
Show that the two formulations are equivalent.

- hartnn

divide by a on both sides?
maybe.....

- anonymous

It's an iff type, so both ways......

- hartnn

idk....maybe multiply both sides by a to get a^p = a (mod p)
but is it correct to do so ?

- anonymous

On the right track, sort of...

- anonymous

a^p = a mod p, then if a not = 0 mod p then can cancel by a to get
a^(p-1) = 1 mod p
That's the first part...

- hartnn

same for other part, right ?

- anonymous

Yes, more or less.....

- anonymous

FLT is also used in some combinatorial problems....

- hartnn

u have some practice/examples with those ?

- anonymous

OK, you can try this one:
Say you have enough beads in n colours, how many different necklaces consisting of p beads can be made, where p is prime?

- hartnn

(mod(n/p))!

- anonymous

I guess I should tell you that you will need to think a bit in order to work it out, it is not straightforward....

- anonymous

Or I can tell you the answer and you can try to work it out.....

- hartnn

let me try....after few minutes, u give answer...

- anonymous

I think u need more than a few minutes....:-)

- hartnn

ok, whats the answer ?

- anonymous

(n^p-n)/2p + (n^((p+1)/2) -n)/2 + n

- hartnn

OMG!

- anonymous

It's an integer so the first term still incorporates FLT...

- anonymous

Do you want links only about FLT/Wilsons or congruences/modular stuff as well....

- hartnn

modular stuff as well...

- anonymous

Similar this http://www.math.sc.edu/~filaseta/gradcourses/Math780notes.pdf
Or more detailed....

- hartnn

thanks, i'll go through it......and will ask u for any doubts.

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