A proof question?

Question

Let n belongs to N and given a belongs to Z, let a* represent the congruence class of a modulo n. if a1* = a2* and b1* = b2* for integers a1,a2,b1,b2, then prove that (a1b1)* = (a2b2)*.

if p is a prime, explain why all elements in Z / pZ have multiplicative inverses.

Answer 1

a1=a2+sn. b1=b2+tn.

Take an element, a1b1, of (a1b1)*.
a1b1=(a2+sn)(b2+tn)
=a2b2+b2sn+a2tn+stnn
=a2b2+(b2s+a2t+stn)n,
which belongs to (a2b2)*.

The second one has to do with the fact that p~0 in Z/pZ, and it has no factors, since it's prime.

Answer 2

a1* = a2* and b1* = b2*
a1-a2=nk, a1=a2+nk
b1-b2=nm, b1=b2+nm
then
a1b1=(a2+nk)(b2+nm)
=a2b2+ a2nm+nkb2+nknm
=a2b2+n(a2 m +kb2+knm)
so a1b1-a2b2=n(a2 m +kb2+knm)
which implies that
(a1b1)* = (a2b2)*

the other question i already answered in the other posting