Hello,
a) We know the binomial coefficients are all integers, so
is an integer.
And we can notice that the numerator p! is divisible by p.
If we take
It means that k! does not contain p, and we can say the same for (p-k)!
So, we have no p at the denominator so the binomial coefficient is divisible by p, meaning this is 0 modulo p.
b) We can write that
We use the result from question a) and the binomial coefficients are 0 modulo p for i=1,2 , ... p-1 so there are only two terms left and then,
c) Let's prove it by induction.
step 1 - for x = 0
This is trivial to notice that
Step 2 - we assume that this is true for k
meaning
and we need to prove that this is true for the k+1
We use the results of b)
and we use the induction hypothesis to say
And it means that this is true for k+1
Step 3 - conclusion
We have just proved by induction the Fermat's little theorem.
p a prime number, for for all x integers
Thank you