proving two statements to be true

I have two questions, and any help would be greatly appreciated.

1. Show that if a = b (mod m) and c = d (mod m), (assume a,b,c,d, and m are all integers with m>=2), then a-c = b-d (mod m).

I started this off by saying that if a=b (mod m), then (a-b)/m = s, where s is some integer, and similarly, that if c=d (mod m), then (d-c)/m = t, where t is some integer. Actually, I don't think this helps any =/.

2. Prove that if n is an odd positive integer, then n^2 = 1(mod 8).

Well, I plugged in odd positive integers for n, and yeah, it's true. But how do I go about proving it?

Thanks guys.