Prove that if gcd(a, m)=1 and gcd(a - 1, m) =1, then 1+a+a²+...+ a² + a(m)−1 = 0 (mod m).

Let f: N11 → N11 be defined by f(x) = (7x + 2) mod 11. Find f-¹ if it exists.

Suppose that j is prime and j2 | ab. Prove that if gcd(a, b) = 1, then j2 | a or p2 | b.

Prove that x3 + 12x2 + 18x + 6 is irreducible in (Z[i])[x].

Let a be a complex number with |a| < 1. Prove that 2 α 1 - āz =1_iff |z| = 1.

Let J5 = {0, 1, 2, 3, 4), and define G: J5 x J5 J5 x J5 as follows. For each (a, b) € J x J₁, G(a, b) = ((3a + 1) mod 5, (2b − 3) mod 5). Find the following. (a) G(2, 2) = (2,0) (b) G(4,1)= (0,4) (c) G(3, 4) (3,3) (d) G(1, 0) (4,2) Need Help? Read It X X X