Prove that gcc n m 1 gcc A km m 1 k NSolution We first pr

Prove that gcc (n, m) = 1 gcc (A + km, m) = 1 k N

Solution

: We first prove this for n = 1 using the binomial theorem. Write (a + b) p = a p + p 1 a p1 b + p 2 a p2 b 2 + . . . + b p However, for 1 m p 1, p m is a multiple of p. To see this, use the definition p m = (p)! m!(pm)! and note p cannot divide m! or (p m)! as all terms in these products are less than p. Therefore all of the terms except the first and last are zero modulo p, so (a + b) p a p + b p (mod p). For general n, use induction (ie (a + b) p 2 (a p + b p ) p a p 2 + b p 2 (mod p))