Congruence Classes

  1. Salutations! I believe I have one implications correct and I am looking for a push in the right direction for the other.

    1. The problem statement, all variables and given/known data
    Let n be an integer and let [itex] [a] \in \thinspace \mathbb{Z}_n [/itex]. Prove that there exists and element [itex] \in \thinspace \mathbb{Z}_n[/itex] such that [itex][a] = 1[/itex] if and only if [itex]\gcd (a,n) = 1[/itex].

    2. The attempt at a solution
    For the [itex](\Longleftarrow) [/itex]case, we know that the [itex]\gcd( a, n ) = 1[/itex] and we are trying to show [a] = [1] in [itex]\mathbb{Z}_n[/itex] We know that [itex] \exists x,y \in \mathbb{Z}[/itex] such that
    [itex] ax + ny = 1 \Longrightarrow ax - 1 = -ny [/itex] but this implies that [itex] a b - 1 = vy[/itex] , where [itex] x = b[/itex] and [itex] v = -n[/itex] and also [itex] v| ab - 1[/itex].

    Now, for the other implications... uh little lost. [a] = [1] implies [ab] = [1]. Can I say that [0] = [n], so [1] = [n+1] = [n] + [1], therefore [ab] - [n] = [1]?

    Thanks in advance!
  2. jcsd
  3. You have [ab] = [1] in Zn. What does that mean in Z?
  4. In Z, it would mean that b is the multiplicative inverse of a, right? In Z we would just have ab = 1
  5. Yes, if ab = 1 in Z, that's right. What I actually wanted was for you to translate this statement as is into Z; that is, tell me what this equivalence class equality means. You've already done this to prove the backwards implication, so you know how. But it's the right place to start.
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