Simple properties of mod operator

Formally prove or disprove (using Division Theorem) the following statements.

1. $ a \pm b \bmod m = (a \bmod m) \pm (b \bmod m) \bmod m $
2. $ a b \bmod m = (a \bmod m) (b \bmod m) \bmod m $
3. $ a^k \bmod m = (a \bmod m)^{(k \bmod m)} \bmod m $
4. $ n \mid m \Longrightarrow (a \bmod m) \bmod n = (a \bmod n) \bmod m $