I am trying to show that the axiom of multiplicative inverses holds on sets of integers modulo P when P is prime.
i just need to show that for any non zero integer, n less than P there is a unique multiple of n such that $nm \mod P = 1$.
i have proven the case that when $n = (p-1)$, m must also be (p-1). since if '$ nm \mod P = 1$', $ nm = px + 1$ (where px is a multiple of p) and $(p-1)^2 = p^2 -2p + 1 = p(p-2) + 1$.
but this only holds for the (p-1) integer and doesn't prove the axiom holds.
I cant figure out how to show that every integer n less than P (not=0) has an m such that nm mod p = 1.
Any help is much appreciated.