You have to prove that nZ is a group under the given sum, not another operation you define. So

i)associativity: since + is associative in Z it must also be associative in any nonempty subset of it, such as nZ.

ii)zero: nZ is the set of elements kn, with k in Z, so 0n=0 also is in nZ

ii)inverse: nZ is the set of elements kn, with k in Z. so if you add to kn, (-k)n, you'll get zero.

b) Do you know anything about cyclic groups? It's realy handy to know cyclic groups when dealing with problems in Z, nZ or Z_n. Anyway, let f:Z -> nZ be such that f(k)=kn. Check that f is an homomorphism and check that it is injective and surjective, and you're done.