Let n ≥ 2 be arbitrary. Then (i) there exists one and only one b ∈ Zn such that b 2 ≡n 0; and (ii) for any a ∈ Zn with a 6= 0, there is not exactly one b ∈ Zn such that b 2 ≡n a. (Hint: think about Exercise 7.81.)
Using paper and pencil (and brute-force calculation), compute the following multiplicative inverses (or state that the inverse doesn’t exist):
Exercise 7.81
Give an algorithm to find the additive inverse of any a ∈ Zn. (Be careful: the additive inverse of a has to be a value from Zn, so you can’t just say that 3’s additive inverse is negative 3
Given your solution to the previous exercise, prove the following properties:
7.82 For any a ∈ Zn, we have −(−a) ≡n a.
7.83 For any a, b ∈ Zn, we have a (−b) ≡n (−a) b.
7.84 For any a, b ∈ Zn, we have a b ≡n (−a) (−b).
In regular arithmetic, for a number x ∈ R, a square root of x is a number y such that y2 = x. If x = 0, there’s only one such y, namely y = 0. If x <> 0, there are two such values y (one positive and one negative). Consider the following claim, and prove or disprove it