# Math Help - Congruences: "The characteristic p binomial theorem"

1. ## Congruences: "The characteristic p binomial theorem"

a. Prove that (a+b)^p is congruent to a^p + b^p (mod p) for any integers a, b, provided p is prime.

b. More generally, prove that if p is prime and q= p^n, where n is a natural number, then (a+b)^q is congruent to a^q + b^q (mod p) for any integers a,b. (Hint: Use Induction.)

$(a+b)^q=\sum\limits_{k=0}^q\binom{q}{k}a^{q-k}{b^k}$ ... and now prove, using induction or whatever, that $q\mid \binom{q}{k}\,\,\,\forall\, \,0 < k< q$ , $q=p^n\,,\,\,p$ a prime.