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### Comparing Corresponding Factors

```Date: 03/04/2003 at 23:40:48
From: John Mark
Subject: Fermat's Theorem

If p is an odd prime and k is an integer satisfying 1 <= k <= p-1,
show that the binomial coefficient

((p-1) k)

is congruent to

(-1)^k(mod p)

I realize that binomial coefficient (p k) is congruent to 0 mod p.
I'm assuming that the statement has to do with a factorization of the
binomial coefficient, but I can not see how to factor it and complete
the proof.
```

```
Date: 03/05/2003 at 03:50:15
From: Doctor Jacques
Subject: Re: Fermat's Theorem

Hi John,

We can write the binomial coefficient C(p-1,k) as:

C(p-1,k) = ((p-1)*...*(p-k))/(1*...*k)

We must show that this is congruent to (-1)^k mod p.

As p is prime, and no factor of the denominator is divisible by p,
this is equivalent to:

(p-1)*...*(p-k) = (-1)^k * (1*...*k) mod p

Notice that both products contain k factors. If you compare
corresponding factors modulo p, this should give you a good hint...

some more, or if you have any other questions.

- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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