Show that if [b] and [c] are both multiplicative inverses of [a] in Z_n then b = c(mod n).

Question

I am having some trouble getting started with this problem. I know that I am going to need the following proposition in the proof: Let n > 0 be an integer. Then the following conditions hold for all integers a, b, c, d: a) If a = c(mod n) and b = d(mod n), then a +/- b = c +/- d (mod n), and ab = cd(mod n). b) If a + c = a + d(mod n), then c = d(mod n). If ac = ad(mod n) and (a,n) = 1, then c = d(mod n).

Any help or suggestions is appreciated!

Answer 1

You have $ab - 1 = kn, ac - 1 = pn\Rightarrow ab - ac = (k-p)n\Rightarrow n\mid a(b-c)$, but $(a,n) = 1 \Rightarrow n \mid (b-c)$