Hello Im working on this problem in my Abstract Algebra clas

Hello. I\'m working on this problem in my Abstract Algebra class and am totally stuck on this question.

Let d, f and n be integers with n > d > 1 also let A = {[k] Zn | [f] [k] = [0]}. Prove if d = gcd(f, n), then the set A contains more than one element and fewer than n elements. [Note that Zn contains exactly n elements: [0], [1], . . . ,[n 1].]

Any help would be much appreciated, thank you.

Solution

Let d, f and n be integers with n > d > 1 then

let A = {[k] Zn | [f] [k] = [0]}.

    here k may be 0 then [f][k]=[0]

      f should not be 0 because d = gcd(f, n) and d>1.

     so that A contain may be 0, other elements also,

so that set A contains more than one element and fewer than n elements(because n is maximum).