I just want to see how well I did with this proof, and would like to know any corrections to make if I did it improperly, thanks!

The proof is Show if a ≡ b(mod m) and n ∈ Z+ where n | m, then a ≡ b(mod n)

My answer:

suppose a ≡ b(mod m). Then n ∈ Z+ , and a,b,m,k ∈ Z. Now suppose a=b+km, then a-b=km and km=n. Thus n|m. Then a ≡ b(mod n). Hence, if a ≡ b(mod m) then a ≡ b(mod n).