Show that in if and only if n is a prime.
(In other words,all coefficients in the difference polynomial are divisible by n. This is a relation in the polynomial ring, not for individual integers x.)
By the binomial formula, you question boils down to: show that is prime if and only if divides for all .
Remember .
Assume is prime. Then it divides the numerator but not the denominator. Since it is prime, we conclude it divides the binomial coefficient. (justify that properly)
Assume is composite. Let be any prime factor of . Of course, . It divides both the largest terms of the numerator and of the denominator of , and it doesn't divide any other factor since there are of them, and only 1 number among consecutive ones is divisible by . Thus, after simplification by , the numerator is not divisible by , so that the binomial coefficient isn't either.