1. Prove that there exists x ∈ Z with ax ≡n 1 if and only if there exists y ∈ Zn with ay ≡n 1. 2. Suppose that the multiplicative inverse a −1 exists in Zn. Let k ∈ Zn be any exponent. Prove that a k...

1. Prove that there exists x ∈ Z with ax ≡n 1 if and only if there exists y ∈ Zn with ay ≡n 1.

2. Suppose that the multiplicative inverse a −1 exists in Zn. Let k ∈ Zn be any exponent. Prove that a k has a multiplicative inverse in Zn, and, in particular, prove that the multiplicative inverse of a k is the kth power of the multiplicative inverse of a. (That is, prove that