Fermat's little Theorem-Help

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In this page:

http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem

In the Second method proof that is given there:

Can someone visit the page and tell me why should it be so that when a,2a,3a,.....(p-1)a when divided by p the remainders are distinct from each other.

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2 Answers

sri 3 ·2010-11-15 05:31:49

What a gap ........ :(

Let us assume the contrary....
Suppose two quantities 'ma' and 'na' leave the same remainder r when divided by p....
ma = cp + r ; na = dp + r [mâ‰ n]
(m-n)a = (c-d)p....
Therefore p divides (m-n)a...
Since a and p are coprime....p divides (m-n)
But this is not possible since m and n are unequal and both are less than p
Q.E.D